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 Floyd–Warshall algorithm compute, and how fast?

  1. AShortest paths from one source vertex to all others in O(V + E)
  2. BShortest paths between all pairs of vertices in O(V³)
  3. CA minimum spanning tree in O(E log V)
  4. DA topological order in O(V²)
2.Algorithms & complexityHard

The Held–Karp dynamic programming algorithm solves the travelling salesman problem exactly. What is its running time for n cities?

  1. AO(2n²)
  2. BO(n!)
  3. CO(n² 2ⁿ)
  4. DO(n³)
3.Data structuresMedium

How many nodes does a perfect binary tree of height h have, with height counted in edges?

  1. A2h+1 − 1
  2. B2h
  3. Ch2
  4. D2h + 1
4.Data structuresHard

Why might a server choose its hash function at random from a universal family?

  1. ATo remove the need to resize the table
  2. BTo make it very unlikely that an attacker can pick many keys that all collide
  3. CTo make the hash table sorted
  4. DTo guarantee that no two keys can ever collide, whatever keys an attacker sends
5.Graph theoryMedium

A connected graph has 6 vertices and 8 edges. How many edges k must be removed to leave a spanning tree?

|V| = 6|E| = 8
k =
6.Graph theoryHard
2327371078ABCDEF

Use Dijkstra’s algorithm to find the length d(F) of the shortest path from A to F in the weighted graph.

start: Aend: F
d(F) =

3Solutions

1.Algorithms & complexity

Answer: B · Shortest paths between all pairs of vertices in O(V³)

Why this is the answer

Floyd–Warshall is a dynamic programming algorithm that allows each vertex in turn as an intermediate point, using O(V³) time.

Answer

B · Shortest paths between all pairs of vertices in O(V³)
2.Algorithms & complexity

Answer: C · O(n² 2ⁿ)

Why this is the answer

There are about n 2ⁿ states, each computed in O(n); this is exponential but far faster than trying all (n − 1)! tours.

Answer

C · O(n² 2ⁿ)
3.Data structures

Answer: A · 2h+1 − 1

Why this is the answer

Summing 1 + 2 + 4 + ... + 2h gives 2h+1 − 1; for example, height 2 gives 7 nodes.

Answer

A · 2h+1 − 1
4.Data structures

Answer: B · To make it very unlikely that an attacker can pick many keys that all collide

Why this is the answer

With universal hashing the expected number of collisions is small for any set of keys, since the attacker does not know which function was chosen.

Answer

B · To make it very unlikely that an attacker can pick many keys that all collide
5.Graph theory

Answer: k = 3

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

6 − 1 = 5

Edges to remove

k = 8 − 5 = 3

Answer

k = 3
6.Graph theory

Answer: d(F) = 13

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

Each line: the vertex settled, its final distance, and the neighbours whose distance improves.

A (0): B: 2, C: 3
B (2): D: 9, E: 10
C (3): E: 6
E (6): F: 13
D (9)
F (13)

Shortest path

A → C → E → F
d(F) = 13

Answer

d(F) = 13

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