Problem description

5.2 - Activity-Selection Problem

• You have a list of $n$ activities you can perform, but all of these activities use the same resource, so you cannot perform two of these activities at the same time
  • Examples:
    • one classroom for multiple classes
    • one doctor, multiple appointments
• You want to do as many activities as you can, which ones should you choose? 

Problem description

5.2 - Activity-Selection Problem

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time

Problem description

5.2 - Activity-Selection Problem

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• In problem-solving, try to generate examples that will invalidate your trivial solutions, so you don't spend too much time working on a wrong approach

Problem description

5.2 - Activity-Selection Problem

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activity

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Problem description

5.2 - Activity-Selection Problem

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Problem description

5.2 - Activity-Selection Problem

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Problem description

5.2 - Activity-Selection Problem

• Let $S = \{a_1, a_2, \dots, a_n\}$ be the set of $n$ activities ordered by their finish times (in increasing order)
• Each activity $a_i$ has start time $s_i$ and finish time $f_i$, $0 \le s_i < f_i$ 
  • $a_i$ takes place during half-open time interval $[s_i,f_i)$
    • $x \in [s_i,f_i)$ means $s_i \le x < f_i$
  • activities $a_i$ and $a_j$ are compatible if $[s_i,f_i)$ and $[s_j,f_j)$ do not overlap
    • i.e. $f_i < s_j$ or $s_i \ge f_j$

Problem description (formal)

5.2 - Activity-Selection Problem

• Input:
  • The set $S = \{a_1, a_2, \dots, a_n\}$ of $n$ activities, sorted in increasing order of finish time
  • The start time $s_i$ and finish time $f_i$ for each activity $a_i$
• ​Output:
  • ​The maximum subset $A = \{a_{\ell_1}, a_{\ell_2}, \dots, a_{\ell_m}\} \subseteq S$ of compatible activities, where two activities $a_i$ and $a_j$ are compatible if $[s_i,f_i)$ and $[s_j,f_j)$ do not overlap, i.e. $f_i < s_j$ or $s_i \ge f_j$

Problems and Subproblems?

5.2 - Activity-Selection Problem

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Problems and Subproblems

5.2 - Activity-Selection Problem

$[  ]$

$[  ]$

pick $a_1$?

pick $a_2$?

pick $a_3$?

Problems and Subproblems

5.2 - Activity-Selection Problem

$\text{select}(S)$

pick $a_1$?

pick $a_2$?

• Let's call the problem, $\text{select}(S)$ where $S$ is a set of activities (remember that we want to return the set of compatible activities)

$\text{select}(S\setminus\{a_1\})$

$\text{select}(S\setminus\{a_1,a_2\})$

$\text{select}(S\setminus\{a_1\})$

$\text{select}(S\setminus\{a_1\})$

$\text{select}(S\setminus\{a_1\})$

$\text{select}(S)$

$\text{select}(S)$

Problems and Subproblems

5.2 - Activity-Selection Problem

• We can do better: if we pick $a_k$, then we cannot pick another activity that runs on the same time, i.e. whatever activity we choose next must either finish before $a_k$ starts or starts after $a_k$ finishes
• Let $S_{i,j}$ be the set of activities that start after activity $a_i$ finishes, and finishes before activity $a_j$ starts

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$a_1$

$a_{11}$

Problems and Subproblems

5.2 - Activity-Selection Problem

• Let $S_{i,j}$ be the set of activities that start after activity $a_i$ finishes, and finishes before activity $a_j$ starts
• So once we pick an activity $a_k$, we can only pick $S_{?,k}$ and $S_{k,?}$
  • hmm, how does this notation work?

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$a_{4}$

Problems and Subproblems

5.2 - Activity-Selection Problem

• Here's the problem, what is the value of $i$ and $j$ for $S_{i,j}$ that denotes all the activities before or after $a_k$,
  • or more simply: how do you denote the set of all activities?
    • we have 11 activities, so $S_{1,11}$ denotes the set of activities that starts after activity 1 finishes and finishes before activity 11 starts
  • solution: make a couple of dummy/phantom activities:
    • $a_0$, finishes before $a_1$ starts
    • $a_{12}$, starts after $a_{11}$ finishes
  • so
    • $S_{0,k}$ denotes all the activities the finishes before $a_k$ starts (but after this phantom activity finishes)
    • $S_{k,12}$ denotes all the activities that starts after $a_k$ finishes (but before this phantom activity starts)
    • $S_{0,12}$ denotes the set of all activities

Problems and Subproblems

5.2 - Activity-Selection Problem

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$a_{4}$

• Make sure you understand the notation, it may be a little unusual, e.g. $S_{0,2}$ is NOT $\{a_1\}$, it's actually empty (because nothing finishes before $a_2$ starts)
• $S_{0,1}$ and $S_{11,12}$ are empty sets

Problems and Subproblems

5.2 - Activity-Selection Problem

$\text{select}(S_{0,12})$

$\text{select}(S_{0,1}) + \text{select}(S_{1,12})$

$\text{select}(S_{0,11}) + \text{select}(S_{11,12})$

$\text{select}(S_{0,2}) + \text{select}(S_{2,12})$

$\text{select}(S_{0,10}) + \text{select}(S_{10,12})$

pick $a_1$

$\dots$

• The search tree can be written more nicely now:

pick $a_2$

pick $a_{10}$

pick $a_{11}$

Problems and Subproblems

5.2 - Activity-Selection Problem

• Of course you can generalise this:
  • if we start with $S_{i,j}$, then if we pick $a_k$, we have the subproblems $S_{i,k}$ and $S_{k,j}$

$\text{select}(S_{i,j})$

$\text{select}(S_{i,i+1}) + \text{select}(S_{i+1,j})$

$\text{select}(S_{i,j-1}) + \text{select}(S_{j-1,j})$

$\text{select}(S_{i,i+2}) + \text{select}(S_{i+2,j})$

$\text{select}(S_{i,k}) + \text{select}(S_{k,j})$

pick $a_{i+1}$

pick $a_{j-1}$

pick $a_{i+2}$

$\dots$

$\dots$

pick $a_{k}$

Problems and Subproblems

5.2 - Activity-Selection Problem

• Finally, let $A_{i,j}$ be the optimal solution $\text{select}(S_{i,j})$, that is, $A_{i,j}$ is the maximum set of compatible activities in $S_{i,j}$

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$a_{4}$

$a_{8}$

$A_{2,11}$

Optimal Substructure

5.2 - Activity-Selection Problem

$\text{select}(S_{i,j})$

$\text{select}(S_{i,k})$

$\text{select}(S_{k,j})$

optimal solution: $A_{i,j}$

optimal solution: $A_{i,k}$

optimal solution: $A_{k,j}$

• i.e. $A_{i,j} = A_{i,k} \cup \{a_k\} \cup A_{k,j}$ and the optimal solution is $|A_{i,j}| = |A_{i,k}| + 1 +  |A_{k,j}|$

Optimal Substructure

5.2 - Activity-Selection Problem

• Suppose there is a better solution for the subproblem $S_{i,k}$, say $A^\prime_{i,k}$
• ​This means that $|A^\prime_{i,k}| > |A_{i,k}|$, so $|A^\prime_{i,k}| + 1 +  |A_{k,j}|  > |A_{i,j}|$
  • this contradicts the assumption that $A_{i,j}$ is the optimal solution to the problem
• ​We can use a symmetric argument for the solution to the subproblem $S_{k,j}$

$\text{select}(S_{i,j})$

$\text{select}(S_{i,k})$

$\text{select}(S_{k,j})$

optimal solution: $A_{i,j}$

solution: $A_{i,k}$

optimal solution: $A_{k,j}$

optimal solution: $A^\prime_{i,k}$

Recursive Relation

5.2 - Activity-Selection Problem

$\text{select}(S_{i,j})$

$\text{select}(S_{i,i+1}) + \text{select}(S_{i+1,j})$

$\text{select}(S_{i,j-1}) + \text{select}(S_{j-1,j})$

$\text{select}(S_{i,i+2}) + \text{select}(S_{i+2,j})$

$\text{select}(S_{i,k}) + \text{select}(S_{k,j})$

pick $a_{i+1}$

pick $a_{j-1}$

pick $a_{i+2}$

$\dots$

$\dots$

pick $a_{k}$

$$|A_{i,j}| = \begin{cases}\displaystyle{\max_{a_k\in S_{i,j}}} \left(|A_{i,k}| + 1 + |A_{k,j}|\right) & \text{if $S_{i,j}$ is not empty}\\             0  & \text{if $S_{i,j}$ is empty} \end{cases}$$

Greedy Choice

5.2 - Activity-Selection Problem

• What is the greedy choice?
  • we want to maximise the number of activities
  • which activity, if chosen, leaves the as much resource as possible so we can fit in more activities?

Greedy Choice

5.2 - Activity-Selection Problem

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• Pick the one that finishes earliest
• Remember that the activities are sorted according to which one finishes first, so $a_1$ will finish before $a_2$

Greedy Choice

5.2 - Activity-Selection Problem

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• Alternatively, pick the one that starts latest

Greedy Choice

5.2 - Activity-Selection Problem

• In dynamic programming, the hard part is in working out what the subproblems are, and then defining the recursive relationship
  • overlapping subproblems are usually easy to spot
  • proof for optimal substructure are all very similar
• In greedy algorithm, the hard part is in working out what is the greedy choice
  • can you think of a bad greedy choice for the activity selection problem?
  • we mentioned this at the start

Greedy Choice

5.2 - Activity-Selection Problem

• Remember the steps (this is the first approach)
  • Show that there is an optimal substructure
  • Show the recursive relation that gives optimal solution
  • Show that if we make the greedy choice, only one subproblem remains
  • Show that it is safe to make the greedy choice
  • Find the value of the optimal solution (run the algorithm)
  • Develop the recursive solution to an iterative one

Greedy Choice

5.2 - Activity-Selection Problem

• Alternatively:
  • Show that it is safe to make a greedy choice such that our problem only has one subproblem to solve
  • Show that the problem has an optimal substructure after we make the greedy choice
  • Find the value of the optimal solution (run the algorithm)
  • Develop the recursive solution to an iterative one

Greedy Choice

5.2 - Activity-Selection Problem

Greedy Choice

5.2 - Activity-Selection Problem

• Intuition:
  • pick the activity that finishes first, i.e. $a_1$
    • recall that the activities are sorted according to finish time, in increasing order
  • suppose $a_1$ is not in the optimal solution
    • there must be another activity, say $a_k$ that is the first activity in the optimal solution
    • $a_k$ finishes later than $a_1$
    • so we can substitute $a_1$ in for $a_k$ and still get the optimal number of activities

Greedy Choice

5.2 - Activity-Selection Problem

• Theorem:
  • If $a_m$ is an activity in $S_{k,j}$ with the earliest finish time, then $a_m$ is in a maximum-size subset of compatible activities of $S_{k,j}$ (i.e. $a_m \in A_{k,j}$) 
• Proof:
  • let $a_\ell$ be the activity in $A_{k,j}$ with the earliest finish time.
  • if $a_\ell = a_m$, then we are done
  • if $a_\ell \ne a_m$, then replace $a_\ell$ with $a_m$
    • i.e. let $A^\prime_{k,j} = A_{k,j} - \{a_\ell\} + \{a_m\}$
  • Since the activities in $A_{k,j}$ are compatible, and $a_\ell$ is the first activity in $A_{k,j}$ to finish and $f_m \le f_\ell$,
    • then it follows that the activities in $A^\prime_{k,j}$ are also compatible
  • Since $|A^\prime_{k,j}| = | A_{k,j}|$
    • it follows that $A^\prime_{k,j}$ is also a maximum-size subset of compatible activities of $S_{k,j}$

Recursive Solution

5.2 - Activity-Selection Problem

Iterative Solution

5.2 - Activity-Selection Problem