Recursion

What is recursion

Recursion in computer science is a method where the solution to a problem depends on solutions to smaller instances of the same problem (as opposed to iteration).

How to solve a problem

  • Divide into small problems.
  • Solve small problems.
  • Use the result of small problems to solve the original problem.
  • If the small problems are the same as original one, just scale is different. Then this is called recursion.

Recursion Properties

  • Base case
    • simple scenario that does not need recursion to produce an answer. 
  • Recursion
    • a set of rules that reduce all other case toward the base case. 

Code problem with recursion

  • Base case
  • recursion rules
  • Represent the problem with coding function.
    • ​Define the essential parameters
      • Parameters that define the problem.
      • Parameters that store the temporary result or state.​
    • ​Define the return value
  • Fibonacci Number
    • Can be solved by DP, which is better
  • Climbing Stairs
    • Same as Fibonacci
  • Merge Sort
    • We will talk more about that in Sort

Classical Recursion Problem

Problem

sub-problem-1

base-problem

sub-problem-2

sub-problem-3

sub-sub-problem-2

sub-sub-sub-problem

sub-sub-problem-1

...

...

...

Problem

sub-problem-1

sub-sub-problem

sub-sub-sub-problem

base-problem

Problem

sub-problem-1

base-problem

sub-problem-2

sub-problem-3

sub-sub-problem-2

sub-sub-sub-problem

sub-sub-problem-1

...

...

...

Problem

sub-problem-1

sub-sub-problem

sub-sub-sub-problem

base-problem

Problem

sub-problem-1

base-problem

sub-problem-2

sub-problem-3

sub-sub-problem-2

sub-sub-sub-problem

sub-sub-problem-1

...

...

...

Problem

sub-problem-1

sub-sub-problem

sub-sub-sub-problem

base-problem

Problem

sub-problem-1

base-problem

sub-problem-2

sub-problem-3

sub-sub-problem-2

sub-sub-sub-problem

sub-sub-problem-1

...

...

...

Problem

sub-problem-1

sub-sub-problem

sub-sub-sub-problem

base-problem

Problem

sub-problem-1

base-problem

sub-problem-2

sub-problem-3

sub-sub-problem-2

sub-sub-sub-problem

sub-sub-problem-1

...

...

...

Problem

sub-problem-1

sub-sub-problem

sub-sub-sub-problem

base-problem

Problem

sub-problem-1

sub-sub-problem

sub-sub-sub-problem

base-problem

Problem

sub-problem-1

base-problem

sub-problem-2

sub-problem-3

sub-sub-problem-2

sub-sub-sub-problem

sub-sub-problem-1

...

...

...

Backtracking

Recursion with one sub problem

Recursion with 1 sub problem does not have to use recursion

We can think about using iteration instead

 

int Factorial(int n， int total) {
    if(n == 1) return 1;
    return n * Factorial(n-1, n * total);
}
int Factorial(int n) {
    int sum = 1;
    for(int i = 1; i <= n; i ++) {
        sum = sum * i;
    }
    return sum;
}
def factorial(n):
    if n == 1:
        return total
    return n * factorial(n-1)
def factorial(n):
    sum = 1
    for i in range(1, n+1):
        sum = sum * i
    return sum

Gray Code

The gray code is a binary numeral system where two successive values differ in only one bit.

Given a non-negative integer n representing the total number of bits in the code, print the sequence of gray code. A gray code sequence must begin with 0.

For example, given n = 2, return [0,1,3,2]

This is a typical problem with only one sub-problem

You need to find the pattern inside of it.

What is gray code: a form of binary that uses a different method of incrementing from one number to the next

Example: 000 001 011 010 110 111 101 100

                  0      1     3      2      6     7     5      4

Gray Code

The gray code is a binary numeral system where two successive values differ in only one bit.

Given a non-negative integer n representing the total number of bits in the code, print the sequence of gray code. A gray code sequence must begin with 0.

For example, given n = 2, return [0,1,3,2]

n=2

0

1

3

2

n=3

0

1

3

2

6

7

5

4

Symmetric

Gray Code

public List<Integer> grayCode(int n) {
    List<Integer> result = new ArrayList<>();
    helper(n, result);
    return result;
}
public void helper(int n, List<Integer> result) {
    if (n == 0) {
       result.add(0);
       return;
    }
    helper(n-1, result);
    int size = result.size();
    int k = 1 << (n - 1);
    for (int i = size - 1; i >= 0; i --) {
        result.add(result.get(i) + k);
    }
    return;
}

The key is to find the relationship between each sub-problem

hold the result

Gray Code

The key is to find the relationship between each sub-problem

def gray_code(n):
    result = []
    helper(n, result)
    return result

def helper(n, result):
    if n == 0:
        result.append(0)
        return
    helper(n-1, result)
    size = len(result)
    k = 1 << (n - 1)
    for i in range(size - 1, -1, -1):
        result.append(result[i] + k)
    return

Gray Code

    def grayCode(self, n: int) -> List[int]:
        if n == 0:
            return [0]
        res = self.grayCode(n - 1)
        # add = 2 ** (n - 1) # this also works
        add = 1 << (n - 1)
        
        for i in range(len(res) - 1, -1, -1):
            res.append(res[i] + add)
        
        return res

Gray Code

public List<Integer> grayCode(int n) {
    List<Integer> result = new ArrayList<>();
    result.add(0);
    for(int i = 0; i < n; i ++) {
        int k = 1 << i;
        int size = result.size();
        for(int j = size - 1; j >= 0; j --) {
            result.add(result.get(j) + k);
        }
    }
    return result;
}

We actually could get rid of the recursion code.

Should we use recursion or iteration?

Same as recursion, why?

Pow(x, n)

Implement pow(x, n), which calculates x raised to the power n (i.e., xn).

 

Input: x = 2.00000, n = 10
Output: 1024.00000

 

Input: x = 2.10000, n = 3

Output: 9.26100

 

Input: x = 2.00000, n = -2
Output: 0.25000
Explanation: 2-2 = 1/22 = 1/4 = 0.25

 

 

Pow(x, n)


    def myPow(self, x: float, n: int) -> float:
        if n == 0:
            return 1
        if n == 1:
            return x

        if n < 0:
            return self.myPow(1 / x, -n)

        if n % 2:
            return self.myPow(x * x, n // 2) * x
        else:
            return self.myPow(x * x, n // 2)

Maze 

Given a maze and a start point and a target point, return whether the target can be reached.

 

Example Input:

Start Point: (0, 0); Target Point (5, 5);

Maze: char[][] = {

   {'.', 'X', '.', '.', '.', 'X'},

   {'.', '.', '.', 'X', '.', 'X'},

   {'X', 'X', '.', 'X', '.', '.'},

   {'.', 'X', 'X', 'X', '.', 'X'},

   {'.', '.', '.', '.', '.', 'X'},

   {'.', '.', '.', '.', '.', '.'}

}

Example Output: True

Maze 

Given a maze and a start point and a target point, return whether the target can be reached.

 

Example Input:

Start Point: (0, 0); Target Point (5, 5);

Maze: char[][] = {

   {'.', 'X', '.', '.', '.', 'X'},

   {'.', '.', '.', 'X', '.', 'X'},

   {'X', 'X', '.', 'X', '.', '.'},

   {'.', 'X', 'X', 'X', '.', 'X'},

   {'.', '.', '.', '.', '.', 'X'},

   {'.', '.', '.', '.', '.', '.'}

}

Example Output: True

Maze 

Given a maze and a start point and a target point, return whether the target can be reached.

Consider what are valid subproblem?

What direction can it go?

  • Out of Bound
  • Wall

Maze 

Given a maze and a start point and a target point, return whether the target can be reached.

  • Out of Bound
  • Wall
  • Visited
    • ​(1, 2) -> (2, 2) -> (1, 2) -> (2, 2) -> ...
    • Why we cannot go visited place?

Maze 

Given a maze and a start point and a target point, return whether the target can be reached.

Maze 

Given a maze and a start point and a target point, return whether the target can be reached.

public static boolean solveMaze(char[][] maze, 
                                int startX, int startY, int targetX, int targetY,
                                boolean[][] visited) {
    if (startX < 0 || startX >= maze.length ||
        startY < 0 || startY >= maze[0].length ||
        maze[startX][startY] == 'X' || visited[startX][startY]) {
      return false;
    }
    if (startX == targetX && startY == targetY) {
      return true;
    }
    visited[startX][startY] = true;
    if (solveMaze(maze, startX + 1, startY, targetX, targetY, visited) ||
        solveMaze(maze, startX, startY + 1, targetX, targetY, visited) ||
        solveMaze(maze, startX - 1, startY, targetX, targetY, visited) ||
        solveMaze(maze, startX, startY - 1, targetX, targetY, visited)) {
      return true;
    }
    return false;
}

Maze 

Given a maze and a start point and a target point, return whether the target can be reached.

Does the base case order matter?

Maze 

Given a maze and a start point and a target point, return whether the target can be reached.

def solve_maze(maze, startX, startY, targetX, targetY, visited):
    if (startX < 0 or startX >= len(maze) or
        startY < 0 or startY >= len(maze[0]) or
        maze[startX][startY] == 'X' or visited[startX][startY]):
        return False
    if startX == targetX and startY == targetY:
        return True
    visited[startX][startY] = True
    if (solve_maze(maze, startX + 1, startY, targetX, targetY, visited) or
        solve_maze(maze, startX, startY + 1, targetX, targetY, visited) or
        solve_maze(maze, startX - 1, startY, targetX, targetY, visited) or
        solve_maze(maze, startX, startY - 1, targetX, targetY, visited)):
        return True
    return False
public static boolean solveMaze(char[][] maze, 
                                int startX, int startY, int targetX, int targetY) {

    if (startX < 0 || startX >= maze.length ||
        startY < 0 || startY >= maze[0].length ||
        maze[startX][startY] == 'X') {
      return false;
    }
    if (startX == targetX && startY == targetY) {
      return true;
    }
    maze[startX][startY] = 'X';
    if (solveMaze(maze, startX + 1, startY, targetX, targetY) ||
        solveMaze(maze, startX, startY + 1, targetX, targetY) ||
        solveMaze(maze, startX - 1, startY, targetX, targetY) ||
        solveMaze(maze, startX, startY - 1, targetX, targetY)) {
      return true;
    }
    return false;
}

Maze 

Given a maze and a start point and a target point, return whether the target can be reached.

Optimization 1: reuse the maze to reduce the space complexity

Maze 

Given a maze and a start point and a target point, return whether the target can be reached.

Optimization 1: reuse the maze to reduce the space complexity

def solve_maze(maze, startX, startY, targetX, targetY):
    if (startX < 0 or startX >= len(maze) or
        startY < 0 or startY >= len(maze[0]) or
        maze[startX][startY] == 'X'):
        return False
    if startX == targetX and startY == targetY:
        return True
    maze[startX][startY] = 'X'
    if (solve_maze(maze, startX + 1, startY, targetX, targetY) or
        solve_maze(maze, startX, startY + 1, targetX, targetY) or
        solve_maze(maze, startX - 1, startY, targetX, targetY) or
        solve_maze(maze, startX, startY - 1, targetX, targetY)):
        return True
    return False
public static boolean solveMaze(char[][] maze, 
                                int startX, int startY, int targetX, int targetY) {

    if (startX < 0 || startX >= maze.length ||
        startY < 0 || startY >= maze[0].length ||
        maze[startX][startY] == 'X') {
      return false;
    }
    if (startX == targetX && startY == targetY) {
      return true;
    }
    maze[startX][startY] = 'X';
    int[] dx = {1, 0, -1, 0};
    int[] dy = {0, 1, 0, -1};
    for (int i = 0; i < 4; i++) {
      if (solveMaze(maze, startX + dx[i], startY + dy[i], targetX, targetY)) {
        return true;
      }
    }
    return false;
}

Maze 

Given a maze and a start point and a target point, return whether the target can be reached.

Optimization 2: code style

Maze 

Given a maze and a start point and a target point, return whether the target can be reached.

Optimization 2: code style

def solve_maze(maze, startX, startY, targetX, targetY):
    if (startX < 0 or startX >= len(maze) or
        startY < 0 or startY >= len(maze[0]) or
        maze[startX][startY] == 'X'):
        return False
    if startX == targetX and startY == targetY:
        return True
    maze[startX][startY] = 'X'
    dx = [1, 0, -1, 0]
    dy = [0, 1, 0, -1]
    for i in range(4):
        if solve_maze(maze, startX + dx[i], startY + dy[i], targetX, targetY):
            return True
    return False
public static boolean solveMaze(char[][] maze, 
                                int startX, int startY, int targetX, int targetY) {

    if (startX == targetX && startY == targetY) {
      return true;
    }
    maze[startX][startY] = 'X';
    int[] dx = {1, 0, -1, 0};
    int[] dy = {0, 1, 0, -1};
    for (int i = 0; i < 4; i++) {
      int newX = startX + dx[i], newY = startY + dy[i];
      if (newX < 0 || newX >= maze.length || newY < 0 || newY >= maze.length ||
          maze[newX][newY] == 'X') {
        continue;
      }
      if (solveMaze(maze, newX, newY, targetX, targetY)) {
        return true;
      }
    }
    return false;
}

Maze 

Given a maze and a start point and a target point, return whether the target can be reached.

Optimization 3: move the base case -> is it good?

Maze 

Given a maze and a start point and a target point, return whether the target can be reached.

Optimization 3: move the base case -> is it good?

def solve_maze(maze, startX, startY, targetX, targetY):
    if startX == targetX and startY == targetY:
        return True
    maze[startX][startY] = 'X'
    dx = [1, 0, -1, 0]
    dy = [0, 1, 0, -1]
    for i in range(4):
        newX, newY = startX + dx[i], startY + dy[i]
        if (newX < 0 or newX >= len(maze) or
            newY < 0 or newY >= len(maze[0]) or
            maze[newX][newY] == 'X'):
            continue
        if solve_maze(maze, newX, newY, targetX, targetY):
            return True
    return False
public static boolean solveMaze(char[][] maze, 
                                int startX, int startY, int targetX, int targetY,
                                String path) {
    if (startX < 0 || startX >= maze.length ||
        startY < 0 || startY >= maze[0].length ||
        maze[startX][startY] == 'X') {
      return false;
    }
    if (startX == targetX && startY == targetY) {
      System.out.println(path);
      return true;
    }
    maze[startX][startY] = 'X';
    int[] dx = {1, 0, -1, 0};
    int[] dy = {0, 1, 0, -1};
    char[] direction = {'D', 'R', 'U', 'L'};
    for (int i = 0; i < 4; i++) {
      String newPath = path + direction[i] + " ";
      if (solveMaze(maze, startX+dx[i], startY+dy[i], targetX, targetY, newPath)) {
        return true;
      }
    }
    return false;
}

Maze 

Given a maze and a start point and a target point, print out the path to reach the target.

Maze 

Given a maze and a start point and a target point, print out the path to reach the target.

def solve_maze(maze, startX, startY, targetX, targetY, path):
    if (startX < 0 or startX >= len(maze) or
        startY < 0 or startY >= len(maze[0]) or
        maze[startX][startY] == 'X'):
        return False
    if startX == targetX and startY == targetY:
        print(path)
        return True
    maze[startX][startY] = 'X'
    dx = [1, 0, -1, 0]
    dy = [0, 1, 0, -1]
    direction = ['D', 'R', 'U', 'L']
    for i in range(4):
        newPath = path + direction[i] + " "
        if solve_maze(maze, startX + dx[i], startY + dy[i], targetX, targetY, newPath):
            return True
    return False
public static boolean solveMaze(char[][] maze, 
                                int startX, int startY, int targetX, int targetY,
                                ArrayList<Character> path) {
    if (startX < 0 || startX >= maze.length ||
        startY < 0 || startY >= maze[0].length ||
        maze[startX][startY] == 'X') {
      return false;
    }
    if (startX == targetX && startY == targetY) {
      return true;
    }
    maze[startX][startY] = 'X';
    int[] dx = {1, 0, -1, 0};
    int[] dy = {0, 1, 0, -1};
    char[] direction = {'D', 'R', 'U', 'L'};
    for (int i = 0; i < 4; i++) {
      path.add(direction[i]);
      if (solveMaze(maze, startX+dx[i], startY+dy[i], targetX, targetY, path)) {
        return true;
      }
      path.remove(path.size()-1);
    }
    return false;
}

Maze 

Given a maze and a start point and a target point, return the path to reach the target.

Maze 

Given a maze and a start point and a target point, return the path to reach the target.

def solve_maze(maze, startX, startY, targetX, targetY, path):
    if (startX < 0 or startX >= len(maze) or
        startY < 0 or startY >= len(maze[0]) or
        maze[startX][startY] == 'X'):
        return False
    if startX == targetX and startY == targetY:
        return True
    maze[startX][startY] = 'X'
    dx = [1, 0, -1, 0]
    dy = [0, 1, 0, -1]
    direction = ['D', 'R', 'U', 'L']
    for i in range(4):
        path.append(direction[i])
        if solve_maze(maze, startX + dx[i], startY + dy[i], targetX, targetY, path):
            return True
        path.pop()
    return False

Maze 

Given a maze and a start point and a target point, print out the path to reach the target.

Given a maze and a start point and a target point, return the path to reach the target.

 

Compare these 2, what is the difference?

 

When we share the field, what should we do?

Importance for backtracking -> keep the subproblem has the same state

Backtracking Summary

  • Backtrack = try, iterate, traverse, etc.
  • Keep trying (in the search space) until
    • Solution is found
    • No more meaningful methods to try (no more search space)
  • Level-N problem -> M * Level-(N-1) subproblem
    • Keep states the same when entering subproblem for shared fields.

Permutation (Leetcode 46)

Given a collection of distinct numbers, return all possible permutations.

For example,
[1,2,3] have the following permutations:
[1,2,3], [1,3,2], [2,1,3], [2,3,1], [3,1,2], and [3,2,1].

public List<List<Integer>> permute(int[] nums) {
        // Implement this method.
 }

 

What is the time complexity?

 

O(n *  n!)

Given a collection of distinct numbers, return all possible permutations.

public List<List<Integer>> permute(int[] num) {
    List<List<Integer>> results = new ArrayList<List<Integer>>();
    permute(results, new ArrayList<Integer>(), num);
    return results;
}
    
public void permute(List<List<Integer>> results,
                    ArrayList<Integer> cur, int[] num) {
    if (cur.size() == num.length) {
        results.add(new ArrayList<Integer>(cur));
        return;
    }
    for (int i = 0; i < num.length; i++) {
        if (cur.contains(num[i])) {
            continue;
        }
        cur.add(num[i]);
        permute(results, cur, num);
        cur.remove(cur.size() - 1);
    }
}

Permutation (Leetcode 46)

Which part is not efficient? -> array contains

Given a collection of distinct numbers, return all possible permutations.

Permutation (Leetcode 46)

Which part is not efficient? -> array contains

def permute(num):
    results = []
    permute_helper(results, [], num)
    return results

def permute_helper(results, cur, num):
    if len(cur) == len(num):
        results.append(list(cur))
        return
    for i in num:
        if i in cur:
            continue
        cur.append(i)
        permute_helper(results, cur, num)
        cur.pop()

public ArrayList<ArrayList<Integer>> permute(int[] num) {
    ArrayList<Integer> numList = new ArrayList<Integer>();
    for (int i = 0; i < num.length; i++)
        numList.add(num[i]);
    return permute(new ArrayList<Integer>(), numList);
}

public ArrayList<ArrayList<Integer>> permute(ArrayList<Integer> cur,
                                             ArrayList<Integer> num) {
    ArrayList<ArrayList<Integer>> results =
         new ArrayList<ArrayList<Integer>>();
    if (num.size() == 0) {
        results.add(cur);
        return results;
    }
    for (int i = 0; i < num.size(); i++) {
        ArrayList<Integer> newCur = new ArrayList<Integer>(cur);
        newCur.add(num.get(i));
        ArrayList<Integer> newNum = new ArrayList<Integer>(num);
        newNum.remove(i);
        results.addAll(permute(newCur, newNum));
    }
    return results;
}

Permutation (Leetcode 46)

What part is not efficient?

copy the array takes time

Permutation (Leetcode 46) 

def permute(num):
    num_list = list(num)
    return permute_helper([], num_list)

def permute_helper(cur, num):
    results = []
    if len(num) == 0:
        results.append(cur)
        return results
    for i in range(len(num)):
        new_cur = cur + [num[i]]
        new_num = num[:i] + num[i+1:]
        results.extend(permute_helper(new_cur, new_num))
    return results

Permutation (Leetcode 46) 
How to debug and learn from other solution

def permute(num):
    num_list = list(num)
    return permute_helper([], num_list)

def permute_helper(cur, num):
	print(cur, num)
    results = []
    if len(num) == 0:
        results.append(cur)
        return results
    for i in range(len(num)):
        new_cur = cur + [num[i]]
        new_num = num[:i] + num[i+1:]
        results.extend(permute_helper(new_cur, new_num))
    return results

([], [2, 6, 8])

([2], [6, 8])

([2, 6], [8])

([2, 6, 8], [])

([2, 8], [6])

([2, 8, 6], [])

([6], [2, 8])

([6, 2], [8])

([6, 2, 8], [])

([6, 8], [2])

([6, 8, 2], [])

([8], [2, 6])

([8, 2], [6])

([8, 2, 6], [])

([8, 6], [2])

([8, 6, 2], [])

 

Permutation II (Leetcode 47)

Given a collection of distinct numbers, return all possible permutations.

public List<List<Integer>> permuteUnique(int[] num) {
    List<List<Integer>> results = new ArrayList<List<Integer>>();
    Arrays.sort(num);
    boolean[] visited = new boolean[num.length];
    for (int i = 0; i < visited.length; i++) {
        visited[i] = false;
    }
    permute(results, new ArrayList<Integer>(), num, visited);
    return results;
}

public void permute(List<List<Integer>> results, List<Integer> cur, int[] num, boolean[] visited) {
    if (cur.size() == num.length) {
        results.add(new ArrayList<Integer>(cur));
        return;
    }
    for (int i = 0; i < num.length; i++) {
        if (visited[i] || (i > 0 && num[i] == num[i-1] && !visited[i-1])) {
            continue;
        }
        visited[i] = true;
        cur.add(num[i]);
        permute(results, cur, num, visited);
        cur.remove(cur.size() - 1);
        visited[i] = false;
    }
    return;
}

Permutation II (Leetcode 47)

Given a collection of distinct numbers, return all possible permutations.

def permuteUnique(num):
    results = []
    num.sort()
    visited = [False] * len(num)
    permute(results, [], num, visited)
    return results

def permute(results, cur, num, visited):
    if len(cur) == len(num):
        results.append(list(cur))
        return
    for i in range(len(num)):
        if visited[i] or (i > 0 and num[i] == num[i - 1] and not visited[i - 1]):
            continue
        visited[i] = True
        cur.append(num[i])
        permute(results, cur, num, visited)
        cur.pop()
        visited[i] = False

Permutation II (Leetcode 47)

Given a collection of distinct numbers, return all possible permutations.

public List<List<Integer>> permuteUnique(int[] num) {
    List<List<Integer>> results = new ArrayList<List<Integer>>();
    List<Integer> nums = new ArrayList<Integer>();
    Arrays.sort(num);
    for (int i = 0; i < num.length; i ++) {
        nums.add(num[i]);
    }
    permute(results, nums, 0);
    return results;
}

public void permute(List<List<Integer>> results, List<Integer> num, int index) {
    if (index == num.size()) {
        results.add(num);
        return;
    }
    for (int i = index; i < num.size(); i++) {
        if (i != index && num.get(index) == num.get(i)) continue;
        swap(num, index, i);
        permute(results, new ArrayList<Integer>(num), index + 1);
    }
}
public void swap(List<Integer> num, int i, int j) {
    int temp = num.get(i);
    num.set(i, num.get(j));
    num.set(j, temp);
}

Permutation II (Leetcode 47)

Given a collection of distinct numbers, return all possible permutations.

def permuteUnique(self, nums):
  results = []
  num = sorted(nums)
  self.permute(results, num, 0)
  return results

def permute(self, results, num, index):
  if index == len(num):
    results.append(num[:])
    return
  for i in range(index, len(num)):
    if i != index and num[index] == num[i]:
      continue
      num[index], num[i] = num[i], num[index]
      self.permute(results, num[:], index + 1)

Combination

Given a collection of distinct numbers, return all possible combinations.

 

For example,
[2, 6, 8] have the following powersets (combination):
[], [2], [6], [8], [2, 6], [2, 8], [6, 8], [2, 6, 8].

 

public List<List<Integer>> combine(int[] nums) {
        // Implement this method.
 }

Combination

Given a collection of distinct numbers, return all possible combinations.

public static List<List<Integer>> combine(int[] nums) {
    List<List<Integer>> results = new ArrayList<>();
    combination(results, nums, 0, new ArrayList<Integer>());
    return results;
}

public static void combination(List<List<Integer>> results, 
                               int[] nums, int index, ArrayList<Integer> items) {
    if (index == nums.length) {
      results.add(items);
      return;
    }
    ArrayList<Integer> newItems1 = new ArrayList<Integer>(items);
    combination(results, nums, index+1, newItems1);
    
    ArrayList<Integer> newItems2 = new ArrayList<Integer>(items);
    newItems2.add(nums[index]);
    combination(results, nums, index+1, newItems2);
  }

Combination

Given a collection of distinct numbers, return all possible combinations.

def combine(nums):
    results = []
    combination_helper(results, nums, 0, [])
    return results

def combination_helper(results, nums, index, items):
    if index == len(nums):
        results.append(list(items))
        return
    combination_helper(results, nums, index + 1, items)
    
    new_items = items + [nums[index]]
    combination_helper(results, nums, index + 1, new_items)

Combination

Given a collection of distinct numbers, return all possible combinations.

public static List<List<Integer>> combine(int[] nums) {
    List<List<Integer>> results = new ArrayList<>();
    combination(results, nums, 0, new ArrayList<Integer>());
    return results;
}

public static void combination(List<List<Integer>> results, 
                               int[] nums, int index, ArrayList<Integer> items) {
    if (index == nums.length) {
      results.add(new ArrayList<Integer>(items));
      return;
    }
    combination(results, nums, index+1, items);
    
    items.add(nums[index]);
    combination(results, nums, index+1, items);
    items.remove(items.size()-1);
  }

Combination

Given a collection of distinct numbers, return all possible combinations.

def combine(nums):
    results = []
    combination_helper(results, nums, 0, [])
    return results

def combination_helper(results, nums, index, items):
    if index == len(nums):
        results.append(list(items))
        return
    combination_helper(results, nums, index + 1, items)
    
    items.append(nums[index])
    combination_helper(results, nums, index + 1, items)
    items.pop()

Combination

Given a collection of distinct numbers, return all possible combinations.

public static List<List<Integer>> combine(int[] nums) {
    List<List<Integer>> results = new ArrayList<>();
    combination(results, nums, 0, new ArrayList<Integer>());
    return results;
}

public static void combination(List<List<Integer>> results, 
                               int[] nums, int index,
                               ArrayList<Integer> items) {
    if (index == nums.length) {
      results.add(new ArrayList<Integer>(items));
      return;
    }
    for (int i = index; i < nums.length; i++) {
      items.add(nums[i]);
      combination(results, nums, i+1, items);
      items.remove(items.size()-1);
    }
}

The empty set is missing.

-There will be at least one element in the "items"

input: 2 6 8

output:

8

2,8

6,8

2,6,8

Combination

Given a collection of distinct numbers, return all possible combinations.

def combine(nums):
    results = []
    combination_helper(results, nums, 0, [])
    return results

def combination_helper(results, nums, index, items):
  results.append(list(items))
  if index == len(nums):
    return
  for i in range(index, len(nums)):
    items.append(nums[i])
    combination_helper(results, nums, i + 1, items)
    items.pop()

Combination

Given a collection of distinct numbers, return all possible combinations.

public List<List<Integer>> combine(int[] nums) {
  List<List<Integer>> results = new ArrayList<>();
  combinationHelper(results, nums, 0, new ArrayList<>());
  return results;
}

private void combinationHelper(List<List<Integer>> results, int[] nums, int index, List<Integer> items) {
  results.add(new ArrayList<>(items));
  if (index == nums.length) {
    return;
  }
  for (int i = index; i < nums.length; i++) {
    items.add(nums[i]);
    combinationHelper(results, nums, i + 1, items);
    items.remove(items.size() - 1);
  }
}

Lucky Numbers

888 is a lucky number. And for each American phone number, we can actually add some operators to make it become 888. For example:

phone number is 7765332111, you will have

7/7*65*3+3*21*11 = 888

776+5+3*32+11*1 = 888

...

We want to get a full list of all the operation equations that can get a certain lucky number. The interface will be

List<String> luckyNumbers(String num, int target)

 

Lucky Numbers

We want to get a full list of all the operation equations that can get a certain lucky number. The interface will be

​List<String> luckyNumbers(String num, int target)

 

Additional information:

String num will always be 10 digits since it is a phone number.

we can have "0" but we cannot have "05", "032".

If a number cannot be divided, you cannot use it. For example, 55/2 is not allowed, you need to make sure the division can always be an integer result.

0 cannot be divided.

 

Lucky Numbers

We need to use recursion to solve the problem

 

Since we want to output the result. There are things we need to pay attention:

1. The first number does not have operators in front.

2. Use long in case you get the number overflow

3. * and / has higher priority than + and -. How do you handle that?

4. remind 0

Lucky Numbers

def lucky_numbers(num, target):
    result = []
    recursion(num, target, "", 0, 0, 0, result)
    return result

def recursion(num, target, temp, pos, current, last, result):
    if pos == len(num):
        if current == target:
            result.append(temp)
        return
    for i in range(pos, len(num)):
        if num[pos] == '0' and i != pos:
            break
        m = num[pos:i + 1]
        n = int(m)
        if pos == 0:
            recursion(num, target, temp + m, i + 1, n, n, result)
        else:
            recursion(num, target, temp + "+" + m, i + 1, current + n, n, result)
            recursion(num, target, temp + "-" + m, i + 1, current - n, -n, result)
            recursion(num, target, temp + "*" + m, i + 1, current - last + last * n, last * n, result)
            if n != 0 and last % n == 0:
                recursion(num, target, temp + "/" + m, i + 1, current - last + last // n, last // n, result)

Lucky Numbers

public List<String> luckyNumbers(String num, int target) {
    List<String> result = new ArrayList<>();
    recursion(num, target, "", 0, 0, 0, result);
    return result;
}
public void recursion(String num, int target, String temp, int pos, 
                        long current, long last, List<String> result) {
    if (pos == num.length()) {
        if (current == target) {
            result.add(temp);
        }
        return;
    }
    for (int i = pos; i < num.length(); i ++) {
        if (num.charAt(pos) == '0' && i != pos) break;
        String m = num.substring(pos, i + 1);
        long n = Long.valueOf(m);
        if (pos == 0) {
            recursion(num, target, temp + m, i + 1, n, n, result);
        } else {
            recursion(num, target, temp + "+" + m, i + 1, current + n, n, result);
            recursion(num, target, temp + "-" + m, i + 1, current - n, -n, result);
            recursion(num, target, temp + "*" + m, i + 1, current - last + last * n, last * n, result);
            if (n != 0 && last % n == 0) {
                recursion(num, target, temp + "/" + m, i + 1, current - last + last / n, last / n, result);
            }
        }
    }
}

Lucky Numbers

Example:

24 + 5 -> current 29, last 5

24 + 5 * 4 -> current 29 - 5 + 20 = 44, last 20

24 + 5 * 4 / 2 -> current 44 - 20 + 10 = 34, last 10

24 + 5 * 4 / 2 - 7 -> current 34 - 7 = 27, last -7

Knapsack

Given a set of candidate numbers (C) and a target number (T), find all unique combinations in C where the candidate numbers sums to T.

All candidate numbers are unique.

The same repeated number may be chosen from C unlimited number of times.

 

Note:

  • All numbers (including target) will be positive integers.
  • Elements in a combination (a1, a2, … , ak) must be in non-descending order. (ie, a1 ≤ a2 ≤ … ≤ ak).
  • The solution set must not contain duplicate combinations.

 

Example Input: [7], [2, 3, 6, 7]

Example Output: [[2, 2, 3], [7]]

Knapsack

Similar as the previous Knapsack, consider the subproblems

Define the problem as (C, i, T), then given a number

  • Pick => (C, i, T-C[i])
  • Not Pick => (C, i+1, T)

 

Why i does not move when we pick?

Given a set of candidate numbers (C) and a target number (T), find all unique combinations in C where the candidate numbers sums to T.

All candidate numbers are unique.

The same repeated number may be chosen from C unlimited number of times.

Knapsack

public static ArrayList<ArrayList<Integer>> knapsack(int[] candidates, 
                                                     int target) {
    ArrayList<ArrayList<Integer>> results = new ArrayList<ArrayList<Integer>>();
    ArrayList<Integer> cur = new ArrayList<Integer>();
    knapsack(candidates, 0, target, results, cur);
    return results;
}
public static void knapsack(int[] candidates, int index, int target,
                            ArrayList<ArrayList<Integer>> results,
                            ArrayList<Integer> cur) {
    if (target < 0 || (target != 0 && index == candidates.length)) {
        return;
    }
    if (target == 0) {
        results.add(new ArrayList<Integer>(cur));
        return;
    }
    cur.add(candidates[index]);
    knapsack(candidates, index, target-candidates[index], results, cur);
    cur.remove(cur.size()-1);
    knapsack(candidates, index+1, target, results, cur);
}

Given a set of candidate numbers (C) and a target number (T), find all unique combinations in C where the candidate numbers sums to T.

why do we copy?

Knapsack

Given a set of candidate numbers (C) and a target number (T), find all unique combinations in C where the candidate numbers sums to T.

def knapsack(candidates, target):
    results = []
    cur = []
    knapsack_helper(candidates, 0, target, results, cur)
    return results

def knapsack_helper(candidates, index, target, results, cur):
    if target < 0 or (target != 0 and index == len(candidates)):
        return
    if target == 0:
        results.append(cur[:])
        return
    cur.append(candidates[index])
    knapsack_helper(candidates, index, target - candidates[index], results, cur)
    cur.pop()
    knapsack_helper(candidates, index + 1, target, results, cur)

Knapsack

Given a set of candidate numbers (C) and a target number (T), find all unique combinations in C where the candidate numbers sums to T.

All candidate numbers are unique.

The same repeated number may be chosen from C unlimited number of times.

Another way to solve the problem?

Recursion does not have to be one way to solve

 

Iterate all numbers to decide whether to pick the current number.

<=>

Which one of the numbers should I pick as the smallest one in the current set.

Knapsack

Given a set of candidate numbers (C) and a target number (T), find all unique combinations in C where the candidate numbers sums to T.

public ArrayList<ArrayList<Integer>> knapsack(int[] candidates, int target) {
    ArrayList<ArrayList<Integer>> results = new ArrayList<ArrayList<Integer>>();
    ArrayList<Integer> cur = new ArrayList<Integer>();
    knapsack(candidates, 0, target, results, cur);
    return results;
}
public void knapsack(int[] candidates, int index, int target,
                     ArrayList<ArrayList<Integer>> results,
                     ArrayList<Integer> cur) {
    if (target < 0) {
        return;
    }
    if (target == 0) {
        results.add(new ArrayList<Integer>(cur));
        return;
    }
    for (int i = index; i < candidates.length; i++) {
        cur.add(candidates[i]);
        knapsack(candidates, i, target-candidates[i], results, cur);
        cur.remove(cur.size()-1);
    }
    return;
}

Knapsack

Given a set of candidate numbers (C) and a target number (T), find all unique combinations in C where the candidate numbers sums to T.

def knapsack(candidates, target):
    results = []
    cur = []
    knapsack_helper(candidates, 0, target, results, cur)
    return results

def knapsack_helper(candidates, index, target, results, cur):
    if target < 0:
        return
    if target == 0:
        results.append(cur[:])
        return
    for i in range(index, len(candidates)):
        cur.append(candidates[i])
        knapsack_helper(candidates, i, target - candidates[i], results, cur)
        cur.pop()
    return

Knapsack II

Given a collection of candidate numbers (C) and a target number (T), find all unique combinations in C where the candidate numbers sums to T.

Candidate numbers may contain duplicate.

Each number in C may only be used once in the combination.

 

Knapsack I

input: [2,3,7], 9 -> [[2,2,2,3],[2,7],[3,3,3]]

Knapsack II

input: [2,2,3,7], 9 -> [[2,7]]

Knapsack II

public List<List<Integer>> knapsack(int[] candidates, int target) {
    Arrays.sort(candidates);
    List<List<Integer>> results = new ArrayList<List<Integer>>();
    List<Integer> cur = new ArrayList<Integer>();
    knapsack(candidates, 0, target, results, cur);
    return results;
}

public void knapsack(int[] candidates, int index, int target,
                     List<List<Integer>> results, List<Integer> cur) {
    if (target < 0)  return;
    if (target == 0) {
        results.add(new ArrayList<Integer>(cur));
        return;
    }
    for (int i = index; i < candidates.length; i++) {
        cur.add(candidates[i]);
        knapsack(candidates, i+1, target-candidates[i], results, cur);
        cur.remove(cur.size()-1);
        while (i < candidates.length-1 && candidates[i] == candidates[i+1]) i++;
    }
    return;
}

when it is used, need to use the next

Why do we need to skip?

-> avoid duplicate subproblem

Knapsack II

def knapsack(candidates, target):
    candidates.sort()
    results = []
    cur = []
    knapsack_helper(candidates, 0, target, results, cur)
    return results

def knapsack_helper(candidates, index, target, results, cur):
    if target < 0:
        return
    if target == 0:
        results.append(cur[:])
        return
    for i in range(index, len(candidates)):
        cur.append(candidates[i])
        knapsack_helper(candidates, i + 1, target - candidates[i], results, cur)
        cur.pop()
        while i < len(candidates) - 1 and candidates[i] == candidates[i + 1]:
            i += 1
    return

0-1 Knapsack II

Given a knapsack which can hold s pounds of items, and a set of items with weight w1, w2, ... wn. Try to put items into the pack as many as possible, return the largest weight we can get in the knapsack.

public int knapsack(int s, int[] weights) {
    return knapsack(s, weights, 0);
}

public int knapsack(int s, int[] weights, int index) {
    if (s == 0 || index == weights.length) {
      return 0;
    }
    if (weights[index] > s) {
      return knapsack(s, weights, index+1);
    }
    return Math.max(knapsack(s, weights, index+1),
                    weights[index] + knapsack(s-weights[index], weights, index+1));
}

The subproblem's relationship has changed.

This is not the best solution, we can use Dynamic Programming

0-1 Knapsack II

Given a knapsack which can hold s pounds of items, and a set of items with weight w1, w2, ... wn. Try to put items into the pack as many as possible, return the largest weight we can get in the knapsack.

The subproblem's relationship has changed.

This is not the best solution, we can use Dynamic Programming

def knapsack(s, weights):
    return knapsack_recursive(s, weights, 0)

def knapsack_recursive(s, weights, index):
    if s == 0 or index == len(weights):
        return 0
    if weights[index] > s:
        return knapsack_recursive(s, weights, index + 1)
    return max(knapsack_recursive(s, weights, index + 1),
               weights[index] + knapsack_recursive(s - weights[index], weights, index + 1))

0-1 Knapsack 

Given a knapsack which can hold s pounds of items, and a set of items with weight w1, w2, ... wn. Return whether we can pick specific items so that their total weight s.

 

Example Input:

s = 20;

w = [14, 8, 7, 5, 3];

Example Output:

True; (8, 7, 5)

We need to think what are the sub problems

Given an item, just two options:

  • pick it, the problem become (s-w[i], w - w[i])
  • not pick it, the problem become (s,  w - w[i])

0-1 Knapsack 

Given a knapsack which can hold s pounds of items, and a set of items with weight w1, w2, ... wn. Return whether we can pick specific items so that their total weight s.

public boolean knapsack(int s, int[] weights) {
    return knapsack(s, weights, 0);
}

public boolean knapsack(int s, int[] weights, int index) {
	
    if (s == 0) {
      return true;
    }
    if (s < 0 || index == weights.length) {
      return false;
    }
    return knapsack(s - weights[index], weights, index+1) ||
      knapsack(s, weights, index+1);
}

add/remove elements in weights is hard --> add an index and move index

Consider the 2 return, does the order matter?

0-1 Knapsack 

def knapsack(s, weights):
    return knapsack_recursive(s, weights, 0)

def knapsack_recursive(s, weights, index):
    if s == 0:
        return True
    if s < 0 or index == len(weights):
        return False
    return knapsack_recursive(s - weights[index], weights, index+1) or knapsack_recursive(s, weights, index+1)

Summary

Recursion

subproblem

-> problem

subproblem-1 ||

subproblem-2 ||

subproblem-N

-> problem

subproblem-1 &&

subproblem-2 &&

subproblem-N

-> problem

Factorial,

Sum of LinkedList,

Remove Linked List Element, etc.

Maze,

Sudoku,

Most DFS problems, etc.

Knapsack

Permutations,

Combinations,

Eight Queen,

kSum, etc.

Summary

  • Recursion is a strategy.
  • Always try to split a problem into sub-problems and solve the sub-problem first.
    • If the solution is the same, then you can call the same method.

Homework

Homework (Optional)

Letter Combinations of a Phone Number

Subsets

N-Queens

Palindrome Partitioning

Homework (Optional)

Subsets II

Sudoku Solver

Restore IP Address

Word Search

Lettercase Permutation

Copy of [GoValley-Jo] Recursion 1

By ZhiTongGuiGu

Copy of [GoValley-Jo] Recursion 1

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