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COMP-311 Final Flashcards

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1
Q

2-3 Trees

A
  • 1 to 2 values in each node
  • 0 to 3 links to other nodes
  • Balanced, non-binary trees
  • All leaves are at the lowest level
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2
Q

Approximately, what is the maximum height of a binary search tree of N nodes?

A

log2 n

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3
Q

The following items are inserted into a B-tree: 3, 6, 5, 2, 4, 7, 1. Which node is the deepest?

A

Question doesn’t make sense. Must specify the degree of the B-Tree. Even then, all leaves are at the bottom so there will be many nodes at the “deepest” level.

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4
Q

The following items are inserted into a 2-3 tree: 3, 6, 5, 2, 4, 7, 1. Which node is the deepest?

A

3, 5
1, 2 | 4 | 6, 7

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5
Q

What operation(s) will require the most time in a balanced binary search tree?

A

Insertion and deletion have to perform rotations, so they will take longer than search, but they’re all still O(log n).

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6
Q

Show the 2-3 tree after inserting each of the following values one at a time: 1, 4, 9

7
3 | 11

A

7
1, 3 | 11

3, 7
1 | 4 | 11

3, 7
1 | 4 | 9, 11

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7
Q

2-3 Tree Insertion Algorithm

A
  • Find the leaf where the item should be inserted
  • If there is only 1 other item in the leaf, put the new item in the leaf
  • Otherwise, split the node into two other nodes with the leftmost and rightmost elements, ejecting the middle element up to the parent node.
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8
Q

Show the following tree after inserting each of the following one at a time: 9, 13, 17

7
3 | 11, 15

A

7, 11
3 | 9 | 15

7, 11
3 | 9 | 13, 15

11
7 | 15
3 | 9 || 13 | 17

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9
Q

Show the 2-3 tree that would be built for the following sentence: “Rather fail with honor than succeed by fraud.”

A

rather

fail, rather

rather
fail | with

rather
fail, honor | with

rather
fail, honor | than, with

rather, than
fail, honor | succeed | with

rather
fail | than
by | honor || succeed | with

rather
fail | than
by | fraud, honor || succeed | with

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10
Q

2-3-4 Trees

A
  • Like 2-3 trees, but with another data element and reference
  • Node can hold 1-3 data elements
  • Node can hold 0-4 references
  • Split full nodes upon descent for insertion - ensures that there is always space in a leaf to insert the new item
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11
Q

2-3-4 Trees continued

A
  • 2-3-4 trees are equivalent to Red-Black trees
  • 2-node is a black node
  • 4-node is a black node with 2 red children
  • 3- node is a black node with a read left child or a black node with a red right child (arbitrary choice)
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12
Q

B-Trees

A
  • The natural extension to 2-3-4 trees
  • A node holds n data elements and n+1 references to other nodes
  • Said to be of degree n
  • Algorithms work the same as on 2-3-4 trees
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13
Q

B-Tree Uses

A
  • Disk structures large databases and indexes
  • Very high order (200 or more) leads to wide but shallow trees
  • Still need to do many comparisons within a node to locate which reference to follow
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14
Q

Selection Sort Algorithm

A
  • Divide array into left (sorted) and right (unsorted) sections
  • Find smallest element in the right section
  • Swap smallest into first right-side index
  • O(n2)
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15
Q

Insertion Sort Algorithm

A
  • Divide array into left (sorted) and right (unsorted) sections
  • Pick next element in right section
  • Find where it fits in the left section
  • O(n2)
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16
Q

Bubble Sort Algorithm

A
  • So long as an exchange takes place
  • Loop over all the elements pairwise
  • If the elements are out of order, then exchange them
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17
Q

Show the progress of each pass of the selection sort for the following array. How many passes are needed? How many comparisons are performed? How many exchanges?

Original
40 | 35 | 80 | 75 | 60 | 90 | 70 | 75

A
  • # passes: 7 (i.e. n-1)
  • # comparisons: 28 (i.e. n * (n-1)/2)
  • # exchanges: 7 (i.e. n-1)

Pass1
35 | 40 | 80 | 75 | 60 | 90 | 70 | 75

Pass 2
35 | 40 | 80 | 75 | 60 | 90 | 70 | 75

Pass 3
35 | 40 | 60 | 75 | 80 | 90 | 70 | 75

Pass 4
35 | 40 | 60 | 70 | 80 | 90 | 75 | 75

Pass 5
35 | 40 | 60 | 70 | 75 | 90 | 80 | 75

Pass 6
35 | 40 | 60 | 70 | 75 | 75 | 80 | 90

Pass 7
35 | 40 | 60 | 70 | 75 | 75 | 80 | 90

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18
Q

Show the progress of each pass of the bubble sort for the following array. How many passes are needed? How many comparisons are performed? How many exchanges?

Original
40 | 35 | 80 | 75 | 60 | 90 | 70 | 75

A
  • # passes: 4 (varies)
  • # comparisons: 22 (varies 7+6+5+4)
  • # exchanges: 9 (varies)

Pass 1
35 | 40 | 75 | 60 | 80 | 70 | 75 | 90

Pass 2
35 | 40 | 60 | 75 | 70 | 75 | 80 | 90

Pass 3
35 | 40 | 60 | 70 | 75 | 75 | 80 | 90

Pass 4
35 | 40 | 60 | 70 | 75 | 75 | 80 | 90

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19
Q

Show the progress of each pass of the insertion sort for the following array. How many passes are needed? How many comparisons are performed? How many shifts?

Original
40 | 35 | 80 | 75 | 60 | 90 | 70 | 75

A
  • # passes: 7 (i.e. n-1)
  • # comparisons: 15 (varies)
  • # exchanges: 13 (varies)

Pass 1
35 | 40 | 80 | 75 | 60 | 90 | 70 | 75
Pass 2
35 | 40 | 80 | 75 | 60 | 90 | 70 | 75

Pass 3
35 | 40 | 75 | 80 | 60 | 90 | 70 | 75

Pass 4
35 | 40 | 60 | 75 | 80 | 90 | 70 | 75

Pass 5
35 | 40 | 60 | 75 | 80 | 90 | 70 | 75

Pass 6
35 | 40 | 60 | 70 | 75 | 80 | 90 | 75

Pass 7
35 | 40 | 60 | 70 | 75 | 75 | 80 | 90

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20
Q

Shell Sort Algorithm

A
  • A better insertion sort
  • Sorts parallel sub-arrays
  • Decreases gap, and sort larger arrays each pass
  • Proven O(n(3/2)) for gaps from 2k-1
  • Probably O(n(5/4)) or even O(n(7/6)) for initial gap of n/2, then subsequent gaps dividing by 2.2
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21
Q

Heap Sort Algorithm

A
  • Naive algorithm
  • Insert values into a heap
  • Repeatedly extract-max
  • Can be done efficiently in-place
  • Convert array to heap
  • For each element: Swap max with last element, Reconvert to a heap
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22
Q

Tree sort algorithm

A
  • Insert values into a binary search tree
  • Perform in-order traversal to get sorted list
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23
Q

Merge Sort Algorithm

A
  • “Divide and conquer” algorithm
  • Splits list in two
  • Sort sub-lists (recursively)
  • Merge sub-lists
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24
Q

Quick Sort Algorithm

A
  • “Divide and conquer” algorithm
  • Select pivot
  • Partition list into:
    • items less than pivot
    • pivot
    • items greater than pivot
  • Sort the items on either side of the pivot (recursively)
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25
Q

Show the progress of each pass of the shell sort for the following array. Show the array after all sorts when the gap is 3 and the final sort when the gap is 1. List the number of comparisons and exchanges required when the gap is 3 then 2 and then 1. Compare this with the number of comparisons and exchanges that would be required for regular insertion sort.

Original
40 | 35 | 80 | 75 | 60 | 90 | 70 | 75

A

Pass 1, Gap 3
40 | 35 | 80 | 70 | 60 | 90 | 75 | 75

Pass 2, Gap 2
40 | 35 | 60 | 70 | 75 | 75 | 80 | 90

Pass 3, Gap 1
35 | 40 | 60 | 70 | 75 | 75 | 80 | 90

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26
Q

Sort the following number using heapsort. What is the efficiency of this algorithm? 55, 50, 10, 40, 80, 90, 60, 100, 70, 80, 20

A

This is “in-place” O(n log n)

55 | 50 | 10 | 40 | 80 | 90 | 60 | 100 | 70 | 80 | 20

55 | 50 | 10 | 100 | 80 | 90 | 60 | 40 | 70 | 80 | 20

55 | 50 | 90 | 100 | 80 | 10 | 60 | 40 | 70 | 80 | 20

55 | 100 | 90 | 70 | 80 | 10 | 60 | 40 | 50 | 80 | 20

100 | 80 | 90 | 70 | 80 | 10 | 60 | 40 | 50 | 55 | 20

90 | 80 | 60 | 70 | 80 | 10 | 20 | 40 | 50 | 55 | 100

80 | 80 | 60 | 70 | 55 | 10 | 20 | 40 | 50 | 90 | 100

80 | 70 | 60 | 50 | 55 | 10 | 20 | 40 | 80 | 90 | 100

70 | 55 | 60 | 50 | 40 | 10 | 20 | 80 | 80 | 90 | 100

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27
Q

Sort the following numbers using merge sort. What is the efficiency of this algorithm? 55, 50, 10, 40, 80, 90, 60, 100, 70, 80, 20

A
  • recursive and fast [O(n log n)] but requires at least 2n space
  • n comparisons to merge output

Original
55 | 50 | 10 | 40 | 80 | 90 | 60 | 100 | 70 | 80 | 20

Split 1
55 | 50 | 10 | 40 | 80 || 90 | 60 | 100 | 70 | 80 | 20

Split 2
55 | 50 || 10 | 40 | 80 || 90 | 60 | 100 || 70 | 80 | 20

Split 3
55 || 50 || 10 || 40 || 80 || 90 || 60 || 100 || 70 || 80 || 20

Merge 1
55 || 50 || 10 || 40 | 80 || 90 || 60 | 100 || 70 || 20 | 80

Merge 2
50 | 55 || 10 | 40 | 80 || 60 | 90 | 100 || 20 | 70 | 80

Merge 3
10 | 40 | 50 | 55 | 80 || 20 | 60 | 70 | 80 | 90 | 100

Merge 4
10 | 20 | 40 | 50 | 55 | 60 | 70 | 80 | 80 | 90 | 100

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28
Q

Sort the following numbers using tree sort. What is the efficiency of this algorithm? 55, 50, 10, 40, 80, 90, 60, 100, 70, 80, 20

A
  • recursive and fast [O(n log n)] if the tree is kept balanced, but requires at least 4n space
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29
Q

Sort the following numbers using quick sort. What is the efficiency of this algorithm? 55, 50, 10, 40, 80, 90, 60, 100, 70, 80, 20

A
  • recursive and fast [O(n log n)] if the pivot is chosen wisely

Original
55 | 50 | 10 | 40 | 80 | 90 | 60 | 100 | 70 | 80 | 20

Pivot 1
20 | 50 | 10 | 40 || 55 || 80 | 90 | 60 | 100 | 70 | 80

Pivot 2
10 || 20 || 50 | 40 || 55 || 80 | 60 | 80 | 70 || 90 || 100

Pivot 3
10 || 20 || 40 || 50 || 55 || 70 | 60 || 80 || 80 || 90 || 100

Pivot 4
10 || 20 || 40 || 50 || 55 || 60 || 70 || 80 || 80 || 90 || 100

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30
Q

Comparison of sorting algorithms

A
  • All O(n2) algorithms are poor. But of these, insertion sort is fastest due to cache hits
  • All O(n log n) algorithms are adequate. But of these, quick sort is fast when choosing the pivot well.
31
Q

What is a graph?

A
  • A graph is an ordered pair (V, E) where V is a set of nodes called vertices, and E is a collection of node paris called edges
  • Analogy: vertices are destinations and edges are roads connecting destinations
32
Q

Directed vs Undirected Graphs

A
  • Undirected graphs: an edge {A, B} implies that there is a connection from A to B and backward from B to A. i.e. the edge is an unordered pair of vertices
  • Directed graphs: the edge is an ordered pair, i.e. (A, B) is a “one-way” street leading only from A to B
33
Q

Draw the undirected graph whose vertex and edge sets are as indicated below:

  • Vertices: {A, B, C, D, E, F, G}
    • Edges: {{A, E}, {B, G}, {B, C}, {B, A}, {C, E}, {G, D}, {E, F}}
A
34
Q

What’s the difference between two graphs?

A

The directed graph is less connected. For example, there is no path that terminates at B.

35
Q

Referring to the following graphs, indentify the following when you see them: cycle, weighted graph, directed graph, undirected graph, connected graph, and unconnected graph.

A
  • A cycle is a loop in a directed graph
  • A weighted graph has weights associated to each edge, which usually signify some form of cost
  • A directed graph has ordered pairs, signifying the direction from one vertice to the other
  • An undirected graph an edge is an unordered pair of vertices, signifying a connection in both directions
  • A connected graph is when there is an edge to each vertice
  • An unconnected graph is where the graph can be separated into two separate pieces, and there are no edges between those pieces
36
Q

Adjecency Matrix

A
  • Vertices numbered [0..(n-1)]
  • 2D array (n * n). Rows represent outgoing edges, columns represent incident edges
  • Element at index [i, j] is a number with 1.0 indicating an edge from i to j and 0.0 to infinity representing no edge.
37
Q

Adjaceny List

A
  • Vertices numbered [0..(n - 1)]
  • Array of length n representing source vertices
  • Linked list of integers stored at each index representing destination vertices
  • Similar in structure to a hash table with chaining
38
Q

What makes graphs different from most of the other data structures that we have discussed?

A

Most general structure. Not used to just hold or order data. Represent relationships between elements.

39
Q

A network of roads and cities: Should it be directed or undirected? Should the edges have weights? If so, what would they mean?

A
  • Undirected
  • Weighted with a “cost” of travel
  • Cost could be miles, time, or even money (tolls)
40
Q

Will an undirected and weighted graph adequately represent all road connections?

A

No, because a directed graph would be required for one-way streets

41
Q

An electronic circuit with junctions and components: Should it be directed or undirected? Should the edges have weights? If so, what would they mean?

A
  • Digital circuits are directional (have a + & - flow), therefore directed
  • Could be weighted, with resistance of the conductor, but unlikely
42
Q

A frienship graph between people: Should it be directed or undirected? Should the edges have weights? If so, what would they mean? If I am your friend, does that mean you are my friend? How could you tell who has the most friends? What is the largest clique?

A
  • Directed graph A -> B means A considers B a friend. Note A -> B does not imply B -> A
  • Could be wieghted with the “strength” of the friendship on a sociological scale (i.e. acquaintance, casual, close, intimate)
  • Most friends: most outgoing or most incident edges on a vertex?
  • Clique: in graph theory, a clique in an undirected graph G, is a set of vertices V such that for every two vertices in V, there exists an edge connecting the two. This is equivalent to saying that the subgraph induced by V is a complete graph. The size of a clique is the number of vertices it contains.
43
Q

What is the space complexity for the adjacency matrix implementation?

A

Space complexity:

  • |V|2 => number of vertices squared
  • Good when the graph is “dense,” i.e. when |E| ~= |V|2
44
Q

What is the space complexity for the adjacency list implementation?

A

Space complexity:

  • |E| + |V|
  • Good when the graph is “sparse,” i.e. when |E| « |V|2
45
Q

The following four operations are used extensively in the implementations graph algorithms:

A
  • Find edge (u, v)
  • Enumerate all edges
  • Enumerate edges emanating from u
  • Enumerate edges incident on v
46
Q

Find the time complexities for each of the operations depending on the implementation (matrix or list) of the graph. Make a chart for comparison.

A
  • Find edge (u, v)
    • Adjacency list: O(|Eu|)
    • Adjacency matrix: O(1)
  • Enumerate all edges
    • Adjacency list: O(|E|)
    • Adjacency matrix: O(|V|2)
  • Enumerate edges emanating from u
    • Adjacency list: O(|Eu|)
    • Adjacency matrix: O(|V|)
  • Enumerate edges incident on v
    • Adjacency list: O(|E|)
    • Adjacency matrix: O(|V|)
47
Q

Given a “maze” representation convert it to a graph and demonstrate an algorithm to find a path through the maze.

A
  • Start, end, dead ends, and junctions of 3 or more corridors become vertices
  • Apply depth first search or breadth first search to locate the exit
  • Observation: the graph is bifrucated
  • Result: graph is unconnected
  • Conclusion: no path from S to E
48
Q

Given the following undirected graph, illustrate how breadth-first search works when starting at vertex A:

  • V: {A, B, C, D, E}
  • E: {{A, B}, {A, D}, {C, E}, {D, E}}
A

Breadth first: Visit starting vertex, then 1-hop vertices, then 2-hop vertices, etc.

Pass 1

Seen: { A }
Visited: { }
Queue: [A]

Pass 2

Seen: { A, B, D }
Visited: { A }
Queue: [B, D]

Pass 3

Seen: { A, B, D }
Visited: { A, B }
Queue: [D]

Pass 4

Seen: { A, B, D, E }
Visited: { A, B, D }
Queue: [E]

Pass 5

Seen: { A, B, D, E, C }
Visited: { A, B, D, E }
Queue: [C]

Pass 6

Seen: { A, B, D, E, C }
Visited: { A, B, D, E, C }
Queue: []

49
Q

Illustrate Dijkstra’s algorithm on the following graph. (Dijkstra’s algorithm determines the least-cost path from a source vertex to every other vertex in the graph).

A

Starting with our initial vertex

S: { 0 }
VS: { 1, 2, 3, 4 }
p: [0, 0, 0, 0, 0]
d: [0, 10, I, 30, 100]

Compare and select the shortest distance. Update distances if shorter distance is found from newly seen vertex, and update predecesor with new vertex.

S: { 0, 1 }
VS: { 2, 3, 4 }
p: [0, 0, 1, 0, 0]
d: [0, 10, 60, 30, 100]

Choose the next shortest distance. Update distances if shorter distance is found from newly seen vertex, and update predecesor with new vertex.

S: { 0, 1, 3 }
VS: { 2, 4 }
p: [0, 0, 3, 0, 3]
d: [0, 10, 50, 30, 90]

Choose the next shortest distance. Update distances if shorter distance is found from newly seen vertex, and predecesor with new vertex.

S: { 0, 1, 3, 2 }
VS: { 4 }
p: [0, 0, 3, 0, 2]
d: [0, 10, 50, 30, 60]

Review the final vertex, and update the distances and predecesors if applicable.

S: { 0, 1, 3, 2 }
VS: { 4 }
p: [0, 0, 3, 0, 2]
d: [0, 10, 50, 30, 60]

Once all vertices have been seen, then the predecesor array will provide the best path to each vertex.

50
Q

Illustrate Prim’s algorithm on the following graph (Prim’s alogrithm finds a minimum spanning tree)

A
  • Start with original graph
  • Will build a new graph
  • min-heap with edges
  • set of vertices
  • set contains everything but starting vertex
  • while set is empty, if edge not in the heap, it’s added
  • then it will take each edge out of the heap to determine if the vertex has been seen
  • then move to the new vertex with the smallest cost
  • if the new vertex has edges to vertices we’ve already seen, none of the edges are added, and review the remaining edges (if no destinct smallest, becomes a random choice based on implementation)
u = 0
s = { 1, 2, 3, 4, 5 }
h = [{0, 2}, {0, 3}, {0, 1}]
t = []
u = 2
s = { 1, 3, 4, 5 }
h = [{3, 5}, {2, 1}, {3, 3}, {0, 3}, {0, 1}, {2, 4}]
t = [{0, 2}]
u = 5
s = { 1, 4, 5 }
h = [{5, 3}, {2, 1}, {2, 3}, {5, 4}, {0, 3}, {0, 1}, {2, 4}]
t = [{0, 2}, {2, 5}]
u = 3
s = { 1, 4, }
h = [{2, 1}, {2, 3}, {5, 4}, {0, 3}, {0, 1}, {2, 4}]
t = [{0, 2}, {2, 5}, {5, 3}]
u = 1
s = { 4, }
h = [{1, 4}, {2, 3}, {5, 4}, {0, 3}, {0, 1}, {2, 4}]
t = [{0, 2}, {2, 5}, {5, 3}, {2, 1}]
u = 4
s = { }
h = [{2, 3}, {5, 4}, {0, 3}, {0, 1}, {2, 4}]
t = [{0, 2}, {2, 5}, {5, 3}, {2, 1}, {1, 4}]

Minimum spanning tree is: [{0, 2}, {2, 5}, {5, 3}, {2, 1}, {1, 4}]

51
Q

Apply DFS, BFS, Prim’s, and Dijkstra’s algorithms to:

A

One possible DFS:

  • (0, 1, 3, 5, 8, 6, 7)

BFS:

  • (0, 5, 3, 1, … )
  • (0, 5, 3, 1, 10, 8, 6, 4, 2, 11, 9, 7)

Prim’s:

  • [{0, 1}, {1, 3}, {1, 4}, {4, 2}, {0, 5}, {5, 8}, {8, 6}, {6, 9}, {9, 7}, {8, 11}, {11, 10}, {9, 12}]

Dijkstra’s

52
Q

Floyd-Warshall Algorithm

A
  • Works like dijkstra’s, but reviews all possible paths at once
  • reviews
    • if d[i, j] > d[i, k] + d[k, j]
  • If the path from i to j is longer than the path from i to k and k to j, then the latter is shorter
  • Sets the new distance, and updates the path
53
Q

Kruskal’s algorithm

A
  • Spanning tree algorithm, like Prim’s
  • Start with a bunch of one vertex graph (each vertex is independent of all the others)
  • Take the cheapest edge in the whole graph to connect two vertices - create a union on the sets, so now the vertices are part of the same little graph
  • Then take the next cheapest edge
  • If an edge is pulled that connects two already connected pieces, it is ignored
  • A bunch of little graphs that keep growing, until it creates one large spanning tree
  • Works fine as long as there’s a union-find method
54
Q

Comparing Kruskal’s and Prim’s algorithms

A

Prim’s good for dense graphs

  • Adjacency matrix: O(|V|2)
  • Adacjency list: O(|E| + |V| log |V|)

Kruskal’s good for sparse graphs

  • Adjacency matrix: O(|V|2 log |E|)
  • Adacjency list: O(|E| log |E|)
55
Q

AVL Trees

A
  • Interested in relative heights of left and right trees (rheight - lheight)
  • “Balanced” is when left and right trees differ in height by no more than 1. Recursively, left and right subtrees are also balanced.
56
Q

Critically unbalanced trees

A

Four cases:
LL, LR, RR, RL

57
Q

Left-Left rebalancing

A
58
Q

Right- Right rebalancing

A
59
Q

Right-Left rebalancing

A

double rotation

60
Q

Left-Right rebalancing

A
61
Q

Red-Black Trees

A
  • Tree nodes have a color, either red or black
  • The root of the tree is black
  • Red nodes may not have red children
  • The number of black nodes on the paths from every leaf to the root must be the same
  • Case one: color new node red - if parent is black, done
  • Case two: if parent and uncle are red, color parent and uncle black, color grandpaernt red. Recurse onto grandparent
  • Case three: rotate so reds on same side (if necessary), color parent black, grandparent red, rotate again
62
Q

Red-black insertion

A
63
Q

If each branch could only split into two parts at any point, how could Suhai keep the branches and fruit from touching the ground?

A

By keeping the tree balanced, Suhai could allow the maximum number of fruits to grow before the tree touched the ground

64
Q

If the room was 10 meters in height, and each branch point could only happen after about a meter branch, roughly how many fruit could the tree produce?

A

Branch ten times, so 1024

65
Q

The text gives an algorithm for a right rotation. Describe the algorithm for a left rotation

A

You can replace right with left everywhere, and it will be the left rotation

66
Q

Show how the final AVL tree for the “The quick brown fox” changes as you insert “apple,” “cat,” and “hat” in that order.

A
67
Q

Build an AVL tree that inserts the integers 30, 40, 15, 25, 90, 80, 70, 85, 15, 72 in the given order

A
68
Q

Build the AVL tree from the sentence “Now is the time for all good men to come to the aid of the party.”

A
69
Q

Insert the numbers 6, 3, and 0 into the tree

A
70
Q

Build the red-black tree from the sentence “Throw your dreams into space like a kite, and you do not know what it will bring back.”

A
71
Q

Suppose that the root of a red-black tree is red. If you make it black, does the tree remain red-black? What about the reverse situation?

A

The root of a red-black tree is always black. Changing it to black might make it a red-black tree. The reverse situation will make the root red, and so it will not be a red-black tree.

72
Q

What is the largest possible number of internal nodes in a red-black tree with black-height k? What is the smallest possible number?

A

Largest = 22k - 1

Smallest = 2k - 1

73
Q

Comparing RB/AVL Trees

A
  • AVL: average height 1.44 log n
    • An insert can trigger log n rotations
  • RB: average height 2 log n
    • An insert triggers constant (amortized) number of rotations or re-colorings
  • Conclusion: AVL trees are shorter, but cost more time to keep that way

Computer Science (2 decks)

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