DSA1 | Week 2
Data Structures: Just Some Structured Data
Sets & Sequential Data
The atomic types have served us well,
but it is now time to build on them.
Say you are a real estate agent &
need to reserve some houses in a new
apartment complex.
The builder can block any contiguous chunk
of N houses for you.
Your list of clients may evolve over time.
Say you are a real estate agent &
need to reserve some houses in a new apartment complex.
Say you are a real estate agent &
need to reserve some houses in a new apartment complex.
Say you are a real estate agent &
need to reserve some houses in a new apartment complex.
Say you are a real estate agent &
need to reserve some houses in a new apartment complex.
⛔️
Say you are a real estate agent &
need to reserve some houses in a new apartment complex.
⛔️
👇
Say you are a real estate agent &
need to reserve some houses in a new apartment complex.
Say you are a real estate agent &
need to reserve some houses in a new apartment complex.
Say you are a real estate agent &
need to reserve some houses in a new apartment complex.
Say you are a real estate agent &
need to reserve some houses in a new apartment complex.
Say you are a real estate agent &
need to reserve some houses in a new apartment complex.
Say you are a real estate agent &
need to reserve some houses in a new apartment complex.
Say you are a real estate agent &
need to reserve some houses in a new apartment complex.
Say you are a real estate agent &
need to reserve some houses in a new apartment complex.
Say you are a real estate agent &
need to reserve some houses in a new apartment complex.
🔑
Say you are a real estate agent &
need to reserve some houses in a new apartment complex.
🔑
Say you are a real estate agent &
need to reserve some houses in a new apartment complex.
🔑
🔑
Say you are a real estate agent &
need to reserve some houses in a new apartment complex.
🔑
🔑
Say you are a real estate agent &
need to reserve some houses in a new apartment complex.
🔑
🔑
🔑
Say you are a real estate agent &
need to reserve some houses in a new apartment complex.
🔑
🔑
🔑
Say you are a real estate agent &
need to reserve some houses in a new apartment complex.
🔑
🔑
🔑
🔑
Arrays
Linked Lists
Looking up the i-th element
Adding a new element at the start
Adding a new element at the end
Removing the i-th element
instant
instant
instant*
\(\approx\) n
\(\approx\) n
\(\approx\) n-i
\(\approx\) i
Adding a new element at the i-th location
\(\approx\) n-i
\(\approx\) min( i,n-i)**
\(\approx\) min( i,n-i)**
* Assuming we have a tail pointer.
** Assuming we have backward pointers.
Why can't everything be fast?
DYNAMIC ARRAYS
Whenever the array overflows, double its size.
Total cost = total #insertions + total #copy operations
Total cost = total #insertions + total #copy operations
\(\leqslant\) N + N = 2N
...since each copy can be associated with a distinct insert,
& the #inserts = N; so #copy operations \(\leqslant\) N
Why can't everything be fast?
DYNAMIC ARRAYS
Whenever the array overflows, double its size.
Graphs
Model relationships between entities.
Graphs
Common graphical representation: points & lines
Graphs: Terminology
Vertices & Edges
Vertices are adjacent if there is an edge between them; and they are the endpoints of said edge.
An edge is incident to its endpoints.
The (out)-neighbors of a vertex X are all the vertices Y such that (X,Y) is an edge.
The degree of a vertex is the number of neighbors it has.
Representing Graphs
1. Adjacency Matrix
2. Adjacency Lists
3. Edge Lists
Three popular styles.
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Adjacency Matrix
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Adjacency Matrix
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Adjacency List
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Adjacency List
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Edge List
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Edge List
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Edge List
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Edge List
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Official Running Times
Brilliant.
When the procedure only needs constant time.
Decent.
When the procedure always wraps up in, and sometimes needs,
time proportional to the maximum degree of the graph.
(n/m)-tolerable.
When the procedure always wraps up in, and sometimes needs,
time proportional to the number of vertices/edges in the graph.
(n/m)-painful.
When the procedure always wraps up in, and sometimes needs,
time proportional to the number of vertices/edges in the graph squared.
| Adj. Matrix | Adj. List | Edge List | |
|---|---|---|---|
| Adding a vertex | |||
| Deleting a vertex | |||
| Adding an edge | |||
| Deleting an edge | |||
| Finding degree(v) | |||
| Check if (u,v) is an edge |
| Adj. Matrix | Adj. List | Edge List | |
|---|---|---|---|
| Adding a vertex | Painful | ||
| Deleting a vertex | Painful | ||
| Adding an edge | Brilliant | ||
| Deleting an edge | Brilliant | ||
| Finding degree(v) | Tolerable | ||
| Check if (u,v) is an edge | Brilliant |
| Adj. Matrix | Adj. List | Edge List | |
|---|---|---|---|
| Adding a vertex | Painful | Tolerable | |
| Deleting a vertex | Painful | Tolerable | |
| Adding an edge | Brilliant | Brilliant | |
| Deleting an edge | Brilliant | Decent | |
| Finding degree(v) | Tolerable | Decent | |
| Check if (u,v) is an edge | Brilliant | Decent |
| Adj. Matrix | Adj. List | Edge List | |
|---|---|---|---|
| Adding a vertex | Painful | Tolerable | Decent |
| Deleting a vertex | Painful | Tolerable | Tolerable |
| Adding an edge | Brilliant | Brilliant | Brilliant |
| Deleting an edge | Brilliant | Decent | Tolerable |
| Finding degree(v) | Tolerable | Decent | Tolerable |
| Check if (u,v) is an edge | Brilliant | Decent | Tolerable |