Data Structures and Algorithms: cram pack

Formula sheets, the key method steps for every topic and 6 practice problems, with full solutions at the back.

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1Formulas and methods

Graph theory

Handshake lemma
Σ deg(v) = 2|E|
Complete graphs
|E(Kn)| = n(n − 1)/2, |E(Km,n)| = m·n
Regular graph
|E| = n·k/2
Tree and forest
|E| = n − 1, |E| = n − c
Leaves of a tree
L = 2 + Σ (deg(v) − 2) over deg(v) ≥ 3
Euler
circuit: all degrees even; path: exactly 2 odd degrees
Hamilton circuits in Kn
(n − 1)!/2
Dijkstra update
d(v) = min(d(v), d(u) + w(u, v))
Walks of length L
(Mᴸ)ij

Method

  1. Write down the degree of every vertex; the degrees add up to twice the number of edges.
  2. For Euler questions, check that the graph is connected and count the vertices with odd degree.
  3. For trees, use n − 1 edges and the handshake lemma.
  4. For Dijkstra, keep a table of tentative distances and settle the smallest unsettled vertex each round.
  5. For a minimum spanning tree, sort the edges by weight and skip every edge that closes a cycle (Kruskal), or grow one tree from a start vertex (Prim).

2Practice problems

1.Algorithms & complexityMedium

What does the recurrence T(n) = 2T(n − 1) + O(1) solve to?

  1. AO(2ⁿ)
  2. BO(n log n)
  3. CO(n²)
  4. DO(n)
2.Algorithms & complexityHard

Is 22n in O(2ⁿ)?

  1. AYes, because both are exponential
  2. BNo, because 22n = 4ⁿ and 4ⁿ / 2ⁿ = 2ⁿ grows without bound
  3. CYes, because constant factors in the exponent are always ignored in Big O
  4. DNo, because 22n grows more slowly than 2ⁿ
3.Data structuresMedium

An application needs fast lookups and also needs to list keys in sorted order and answer range queries. Which structure fits best?

  1. AAn unsorted linked list
  2. BA balanced binary search tree
  3. CA hash table
  4. DA stack
4.Data structuresHard

A Bloom filter can answer "is this element in the set?" with:

  1. AAnswers only for numbers
  2. BPossible false positives but no false negatives
  3. CExact answers always
  4. DPossible false negatives but no false positives
5.Graph theoryMedium

A connected graph has 9 vertices and 15 edges. How many edges k must be removed to leave a spanning tree?

|V| = 9|E| = 15
k =
6.Graph theoryHard
411111414198ABCDEF

Dijkstra’s algorithm runs from A. Click the vertex that is settled (made permanent) in round 3; A itself is round 1.

start: Around 3
  1. 1A
  2. 2B
  3. 3C
  4. 4D
  5. 5E
  6. 6F

3Solutions

1.Algorithms & complexity

Answer: A · O(2ⁿ)

Why this is the answer

The number of calls doubles with each level and there are n levels, so the total is about 2ⁿ, as in the Towers of Hanoi.

Answer

A · O(2ⁿ)
2.Algorithms & complexity

Answer: B · No, because 22n = 4ⁿ and 4ⁿ / 2ⁿ = 2ⁿ grows without bound

Why this is the answer

A constant added to the exponent gives a constant factor, but a constant multiplying the exponent changes the base and the growth rate.

Answer

B · No, because 22n = 4ⁿ and 4ⁿ / 2ⁿ = 2ⁿ grows without bound
3.Data structures

Answer: B · A balanced binary search tree

Why this is the answer

A balanced BST gives O(log n) lookups and keeps keys ordered, so in-order traversal and range queries are easy; a hash table scatters keys.

Answer

B · A balanced binary search tree
4.Data structures

Answer: B · Possible false positives but no false negatives

Why this is the answer

If any bit for an item is 0, it is definitely absent; if all are 1 it is probably present. Bloom filters trade accuracy for very little memory.

Answer

B · Possible false positives but no false negatives
5.Graph theory

Answer: k = 7

Euler, Hamilton and trees: a connected graph has an Euler circuit (every edge once, back to the start) when every vertex has even degree, and an Euler path that is not a circuit when exactly two vertices have odd degree; the path then runs from one odd vertex to the other. A Hamilton circuit visits every vertex once; Kn has (n − 1)!/2 of them. A tree on n vertices has n − 1 edges, and a connected graph with m edges keeps a spanning tree after removing m − n + 1 edges.

Edges of a spanning tree

9 − 1 = 8

Edges to remove

k = 15 − 8 = 7

Answer

k = 7
6.Graph theory

Answer: C

Shortest paths and spanning trees: Dijkstra’s algorithm settles one vertex per round: the unsettled vertex with the smallest tentative distance. Its distance is then final, and every neighbour v gets d(v) = min(d(v), d(u) + w(u, v)). Kruskal’s algorithm builds a minimum spanning tree by taking the edges from light to heavy and skipping any edge that would close a cycle; Prim’s algorithm grows one tree from a start vertex by always adding the lightest edge that leaves it. A spanning tree of n vertices has n − 1 edges.

Dijkstra round by round

1. A (0): B: 4, C: 11
2. B (4): D: 18
3. C (11): E: 25
4. D (18): E: 19, F: 27
5. E (19)
6. F (27)

round 3

C

Answer

C

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