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

How is the travelling salesman problem classified?

  1. AIt is undecidable
  2. BThe decision version is NP-complete and the optimisation version is NP-hard
  3. CThe decision version is in P but the optimisation version is NP-complete
  4. DIt is in P because Dijkstra's algorithm solves it by visiting every city once
3.Data structuresMedium

Why can a binary heap be stored in an array without any pointers?

  1. AIt is a complete binary tree, so positions follow simple index formulas
  2. BArrays automatically keep the keys in sorted order, so no pointers are needed
  3. CAll keys in a heap are equal
  4. DHeaps never contain more than 100 elements
4.Data structuresHard

What guarantee do the red-black tree rules give about the height of a tree with n keys?

  1. AThe height is at most 2 log2(n + 1)
  2. BThe height is at most n / 2
  3. CThe height is exactly log2 n
  4. DThe height is at most log2 n + 1
5.Graph theoryMedium

A tree has 4 vertices of degree 4, and all its other vertices are leaves (degree 1). How many leaves L does it have?

4 × deg 4
L =
6.Graph theoryHard
1675139134ABCDEF

Dijkstra’s algorithm runs from A. Click the vertex that is settled (made permanent) in round 5; A itself is round 1.

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

3Solutions

1.Algorithms & complexity

Answer: A · 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

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

Answer: B · The decision version is NP-complete and the optimisation version is NP-hard

Why this is the answer

"Is there a tour of length at most k?" is NP-complete; finding the shortest tour is NP-hard, since it is at least as hard.

Answer

B · The decision version is NP-complete and the optimisation version is NP-hard
3.Data structures

Answer: A · It is a complete binary tree, so positions follow simple index formulas

Why this is the answer

Completeness means there are no gaps, so the children of index i are always at 2i + 1 and 2i + 2.

Answer

A · It is a complete binary tree, so positions follow simple index formulas
4.Data structures

Answer: A · The height is at most 2 log2(n + 1)

Why this is the answer

Every path has the same number of black nodes and red nodes cannot be adjacent, so the longest path is at most twice the shortest, giving height O(log n).

Answer

A · The height is at most 2 log2(n + 1)
5.Graph theory

Answer: L = 10

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.

Handshake lemma for a tree

L + 16 = 2(L + 4) − 2
L = 10

Answer

L = 10

Shortcut: L = 2 + Σ (d − 2) over the vertices with degree d ≥ 3.

6.Graph theory

Answer: D

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: 1
2. B (1): C: 7, D: 8, E: 5
3. E (5): F: 18
4. C (7)
5. D (8): F: 17
6. F (17)

round 5

D

Answer

D

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