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 amortised analysis measure?

  1. AThe cost of the single most expensive operation anywhere in the sequence
  2. BThe expected cost over random inputs
  3. CThe average cost per operation over a worst-case sequence of operations
  4. DThe cost of an algorithm on the best input
2.Algorithms & complexityHard

What did the Cook–Levin theorem establish?

  1. AP is not equal to NP
  2. BThe Boolean satisfiability problem (SAT) is NP-complete
  3. CThe halting problem is undecidable
  4. DEvery NP problem can be solved in exponential time
3.Data structuresMedium

An adjacency matrix for a graph with V vertices uses how much memory?

  1. AO(log V)
  2. BO(V²)
  3. CO(E)
  4. DO(V + E)
4.Data structuresHard

What is the main advantage of a skip list over a sorted linked list?

  1. AGuaranteed O(1) search because every node stores a pointer to every other node
  2. BIt stores keys without order
  3. CExpected O(log n) search by using several levels of forward pointers
  4. DIt uses no extra memory
5.Graph theoryMedium

How many different Hamilton circuits H does the complete graph K7 have? Circuits that differ only in the starting vertex or the direction count as the same.

K7
H =
6.Graph theoryHard
6461196112ABCDEF

Dijkstra’s algorithm runs from A. Click the vertex that is settled (made permanent) in round 4; A itself is round 1.

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

3Solutions

1.Algorithms & complexity

Answer: C · The average cost per operation over a worst-case sequence of operations

Why this is the answer

Amortised analysis spreads the cost of rare expensive operations over many cheap ones, giving a guaranteed average for any sequence.

Answer

C · The average cost per operation over a worst-case sequence of operations
2.Algorithms & complexity

Answer: B · The Boolean satisfiability problem (SAT) is NP-complete

Why this is the answer

Cook and Levin showed independently that every problem in NP reduces to SAT in polynomial time, so SAT was the first known NP-complete problem.

Answer

B · The Boolean satisfiability problem (SAT) is NP-complete
3.Data structures

Answer: B · O(V²)

Why this is the answer

An adjacency matrix stores V × V entries, efficient for dense graphs; adjacency lists use O(V + E), better for sparse graphs.

Answer

B · O(V²)
4.Data structures

Answer: C · Expected O(log n) search by using several levels of forward pointers

Why this is the answer

Randomised levels allow searches to skip many nodes, giving expected O(log n) search, insert and delete.

Answer

C · Expected O(log n) search by using several levels of forward pointers
5.Graph theory

Answer: H = 360

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.

Count

H = (7 − 1)!/2 = 720/2 = 360

Answer

H = 360
6.Graph theory

Answer: E

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: 6, C: 4
2. C (4): E: 15
3. B (6): D: 12, E: 8
4. E (8): F: 19
5. D (12): F: 18
6. F (18)

round 4

E

Answer

E

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