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

Which statement about P and NP is correct?

  1. AIt has been proven that P ≠ NP
  2. BNP problems cannot be verified quickly
  3. CEvery problem in P is also in NP
  4. DNP means "not polynomial"
3.Data structuresMedium

What is the cost of inserting an element in the middle of an array of n elements?

  1. AO(n²), because each element moves n times
  2. BO(log n), because of binary search
  3. CO(1), because the index is known
  4. DO(n), because later elements must shift
4.Data structuresHard

What does a splay tree do after each access?

  1. ARecolours the nodes on the search path
  2. BRotates the accessed node up to the root
  3. CRebuilds the whole tree in balanced form
  4. DMoves the accessed node to a leaf
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| = 8
  1. 1Euler circuit
  2. 2Euler path, no circuit
  3. 3neither
6.Graph theoryHard
411118141371ABCDEF

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 · O(n³)

Why this is the answer

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

Answer

A · O(n³)
2.Algorithms & complexity

Answer: C · Every problem in P is also in NP

Why this is the answer

P ⊆ NP: if you can solve a problem quickly you can verify a solution quickly. Whether P = NP is still open.

Answer

C · Every problem in P is also in NP
3.Data structures

Answer: D · O(n), because later elements must shift

Why this is the answer

All elements after the insertion point move one position to the right, which takes time proportional to n.

Answer

D · O(n), because later elements must shift
4.Data structures

Answer: B · 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

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

Answer: Euler path, no circuit

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) = 2, deg(B) = 4, deg(C) = 3, deg(D) = 2, deg(E) = 4, deg(F) = 1

Odd vertices

C, F → 2
Euler path, no circuit

Answer

Euler path, no circuit

The Euler path starts at one odd vertex and ends at the other.

6.Graph theory

Answer: F

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): C: 4
2. C (4): B: 15, E: 12
3. E (12): D: 26, F: 19, B: 13
4. B (13): D: 24
5. F (19)
6. D (24)

round 5

F

Answer

F

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