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 is the time complexity of multiplying two n × n matrices with the standard triple loop?

  1. AO(n²)
  2. BO(n)
  3. CO(n³)
  4. DO(n² log n)
2.Algorithms & complexityHard

Which definition of the class P is correct?

  1. ADecision problems solvable by a deterministic algorithm in polynomial time
  2. BProblems that need exponential time on every computer
  3. CProblems that can be solved in parallel
  4. DProblems whose solutions can be checked but never found in polynomial time
3.Data structuresMedium

A trie (prefix tree) is especially efficient for:

  1. AFinding shortest paths
  2. BPrefix searches such as autocomplete
  3. CSorting numbers
  4. DStoring matrices
4.Data structuresHard

What does a splay tree do after each access?

  1. ARecolours the nodes on the search path
  2. BRebuilds the whole tree in balanced form
  3. CMoves the accessed node to a leaf
  4. DRotates the accessed node up to the root
5.Graph theoryMedium
ABCDEFEuler circuitEuler path, no circuitneither

Does the connected graph have an Euler circuit, an Euler path that is not a circuit, or neither? Click your answer.

|V| = 6|E| = 6
  1. 1Euler circuit
  2. 2Euler path, no circuit
  3. 3neither
6.Graph theoryHard
ABCDEF

How many closed walks of length 2 start and end at D? A walk may repeat vertices and edges. Call the number N.

M = adjacency matrixL = 2
N =

3Solutions

1.Algorithms & complexity

Answer: C · O(n³)

Why this is the answer

There are n² entries and each needs n multiplications, giving n³ operations.

Answer

C · O(n³)
2.Algorithms & complexity

Answer: A · Decision problems solvable by a deterministic algorithm in polynomial time

Why this is the answer

P contains decision problems with a deterministic polynomial-time algorithm, such as checking whether a graph is connected.

Answer

A · Decision problems solvable by a deterministic algorithm in polynomial time
3.Data structures

Answer: B · Prefix searches such as autocomplete

Why this is the answer

Tries store strings by characters along paths, so lookups and prefix queries take time proportional to the key length.

Answer

B · Prefix searches such as autocomplete
4.Data structures

Answer: D · Rotates the accessed node up to the root

Why this is the answer

Splaying moves the accessed node to the root through rotations; a single operation can cost O(n), but the amortised cost is O(log n).

Answer

D · Rotates the accessed node up to the root
5.Graph theory

Answer: neither

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.

Degrees

deg(A) = 1, deg(B) = 3, deg(C) = 2, deg(D) = 2, deg(E) = 3, deg(F) = 1

Odd vertices

A, B, E, F → 4
neither

Answer

neither
6.Graph theory

Answer: N = 2

Degrees: the degree of a vertex is the number of edges that meet it. Every edge has two ends, so the degrees add up to twice the number of edges: Σ deg(v) = 2|E| (the handshake lemma). The number of vertices with odd degree is therefore even. Kn has n(n − 1)/2 edges, a k-regular graph on n vertices has nk/2, and a tree on n vertices has n − 1 edges.

Adjacency matrix

A: 0 1 1 0 0 0
B: 1 0 1 1 1 0
C: 1 1 0 0 0 0
D: 0 1 0 0 1 0
E: 0 1 0 1 0 1
F: 0 0 0 0 1 0

Common neighbours

(M²)DD = 0·0 + 1·1 + 0·0 + 0·0 + 1·1 + 0·0 = 2

Answer

N = 2

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