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

How does Prim's algorithm build a minimum spanning tree?

  1. AIt removes the most expensive edge until no cycles remain
  2. BIt grows one tree from a start vertex, adding the cheapest edge that leaves the tree
  3. CIt runs breadth-first search and keeps the BFS tree
  4. DIt sorts all edges by weight and adds each one unless it would form a cycle in the forest
2.Algorithms & complexityHard

Which definition of the class P is correct?

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

In a binary heap stored in an array starting at index 0, where is the parent of the node at index i?

  1. AAt index ⌊(i − 1) / 2⌋
  2. BAt index 2i + 1
  3. CAt index ⌊i / 2⌋ + 1
  4. DAt index i − 1
4.Data structuresHard

Under simple uniform hashing, what is the expected time of a search in a hash table with chaining, n keys and m buckets?

  1. AO(log n)
  2. BO(n)
  3. CO(1 + α), where α = n / m
  4. DO(m)
5.Graph theoryMedium

How many different Hamilton circuits H does the complete graph K5 have? Circuits that differ only in the starting vertex or the direction count as the same.

K5
H =
6.Graph theoryHard
71261011415999ABCDEFG

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 · It grows one tree from a start vertex, adding the cheapest edge that leaves the tree

Why this is the answer

Prim's algorithm, with a priority queue of crossing edges, runs in O(E log V) with a binary heap.

Answer

B · It grows one tree from a start vertex, adding the cheapest edge that leaves the tree
2.Algorithms & complexity

Answer: B · 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

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

Answer: A · At index ⌊(i − 1) / 2⌋

Why this is the answer

Children of i are at 2i + 1 and 2i + 2, so the parent of i is ⌊(i − 1) / 2⌋; with 1-based indexing it is ⌊i / 2⌋.

Answer

A · At index ⌊(i − 1) / 2⌋
4.Data structures

Answer: C · O(1 + α), where α = n / m

Why this is the answer

Computing the hash costs O(1) and the expected chain length is α, so keeping α bounded by resizing gives O(1) expected time.

Answer

C · O(1 + α), where α = n / m
5.Graph theory

Answer: H = 12

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 = (5 − 1)!/2 = 24/2 = 12

Answer

H = 12
6.Graph theory

Answer: d(F) = 27

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: 7
B (7): C: 19, D: 13
D (13): E: 14, F: 27, G: 22
E (14)
C (19)
G (22)
F (27)

Shortest path

A → B → D → F
d(F) = 27

Answer

d(F) = 27

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