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

Why does any comparison-based search in a sorted array need Ω(log n) comparisons in the worst case?

  1. ABinary search is the only possible algorithm, and it needs log₂ n comparisons
  2. BComparisons take logarithmic time each
  3. CEvery element must be compared at least once
  4. DA decision tree with n possible outcomes must have depth at least log₂ n
3.Data structuresMedium

How does extract-min work in a binary min-heap?

  1. ARemove the last element of the array
  2. BMove the last element to the root and sift it down
  3. CSearch all leaves for the new minimum
  4. DRemove the root and shift all elements one place left
4.Data structuresHard

Kahn's algorithm for topological sorting repeatedly removes vertices with in-degree 0. Which structure holds those vertices?

  1. AA stack of all visited edges
  2. BA queue (or any collection) of vertices whose in-degree is 0
  3. CAn adjacency matrix recording which vertices have already been removed
  4. DA max-heap ordered by out-degree
5.Graph theoryMedium

A connected graph has 8 vertices and 15 edges. How many edges k must be removed to leave a spanning tree?

|V| = 8|E| = 15
k =
6.Graph theoryHard
1143851913ABCDEF

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: B · 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

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

Answer: D · A decision tree with n possible outcomes must have depth at least log₂ n

Why this is the answer

Each comparison at most halves the set of remaining possibilities, so distinguishing n positions needs at least log₂ n comparisons.

Answer

D · A decision tree with n possible outcomes must have depth at least log₂ n
3.Data structures

Answer: B · Move the last element to the root and sift it down

Why this is the answer

The last element fills the hole at the root and is swapped with its smaller child until the heap property holds, taking O(log n).

Answer

B · Move the last element to the root and sift it down
4.Data structures

Answer: B · A queue (or any collection) of vertices whose in-degree is 0

Why this is the answer

Ready vertices are kept in a queue; removing one lowers its neighbours' in-degrees, and any that reach 0 are added, giving O(V + E) time.

Answer

B · A queue (or any collection) of vertices whose in-degree is 0
5.Graph theory

Answer: k = 8

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.

Edges of a spanning tree

8 − 1 = 7

Edges to remove

k = 15 − 7 = 8

Answer

k = 8
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: 11, C: 4
2. C (4): B: 7, E: 9
3. B (7): D: 15
4. E (9): D: 10, F: 22
5. D (10): F: 19
6. F (19)

round 4

E

Answer

E

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