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

Using the master theorem, T(n) = 8T(n/2) + O(n²) solves to:

  1. AΘ(n² log n)
  2. BΘ(n³)
  3. CΘ(n log n)
  4. DΘ(n²)
3.Data structuresMedium

How long does a search for a word of length L take in a trie with a fixed alphabet?

  1. AO(n), where n is the number of words
  2. BO(log n), where n is the number of words
  3. CO(L), independent of how many words are stored
  4. DO(L × n)
4.Data structuresHard

For each element of an array you want the next greater element to its right, in O(n) total time. Which structure helps?

  1. AA binary search tree rebuilt for each element
  2. BA queue that holds every pair of elements
  3. CA stack that keeps elements waiting for their answer
  4. DA hash table of all elements
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
1122101215159613ABCDEFG

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: C · 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

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

Answer: B · Θ(n³)

Why this is the answer

log₂ 8 = 3, and n² grows slower than n³, so the leaves dominate: Θ(n³).

Answer

B · Θ(n³)
3.Data structures

Answer: C · O(L), independent of how many words are stored

Why this is the answer

Each character selects one child, so a lookup takes L steps no matter how many words the trie contains.

Answer

C · O(L), independent of how many words are stored
4.Data structures

Answer: C · A stack that keeps elements waiting for their answer

Why this is the answer

Scanning left to right, pop every stacked element smaller than the current one (the current element is their answer), then push it; each element moves at most twice.

Answer

C · A stack that keeps elements waiting for their answer
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) = 29

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: 11, C: 2
C (2): B: 4, E: 14, G: 11
B (4): D: 14
G (11)
D (14)
E (14): F: 29
F (29)

Shortest path

A → C → E → F
d(F) = 29

Answer

d(F) = 29

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