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

What is the amortised cost of incrementing a binary counter, given that one increment may flip many bits?

  1. AO(n) per increment
  2. BO(1) per increment
  3. CO(n log n) per increment
  4. DO(log n) per increment
3.Data structuresMedium

What is true of the adjacency matrix of an undirected graph?

  1. AIt is symmetric about the main diagonal
  2. BEach row has exactly one 1
  3. CIts main diagonal always contains ones
  4. DIt is always upper triangular
4.Data structuresHard

The time complexity of decrease-key in a binary heap (given the element's position) is:

  1. AO(n)
  2. BO(1)
  3. CO(log n)
  4. DO(n log n)
5.Graph theoryMedium
ABCDEEuler 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| = 5|E| = 6
  1. 1Euler circuit
  2. 2Euler path, no circuit
  3. 3neither
6.Graph theoryHard
47161363195188ABCDEFG

Kruskal’s algorithm builds a minimum spanning tree. Click the edge it adds as edge number 5; skipped edges do not count.

k = 5
  1. 1edge AB
  2. 2edge AC
  3. 3edge BC
  4. 4edge BD
  5. 5edge CE
  6. 6edge DE
  7. 7edge DF
  8. 8edge GC
  9. 9edge GD
  10. 10edge GE

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 · O(1) per increment

Why this is the answer

Over n increments, bit i flips about n / 2ⁱ times, so the total is less than 2n flips, an average of O(1) each.

Answer

B · O(1) per increment
3.Data structures

Answer: A · It is symmetric about the main diagonal

Why this is the answer

In an undirected graph edge {u, v} means both M[u][v] and M[v][u] are set, so the matrix equals its transpose.

Answer

A · It is symmetric about the main diagonal
4.Data structures

Answer: C · O(log n)

Why this is the answer

After decreasing the key, the element sifts up at most the height of the heap. Fibonacci heaps achieve O(1) amortised decrease-key.

Answer

C · O(log n)
5.Graph theory

Answer: neither

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) = 3, deg(C) = 3, deg(D) = 1, deg(E) = 3

Odd vertices

B, C, D, E → 4
neither

Answer

neither
6.Graph theory

Answer: edge AC

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.

Kruskal’s algorithm

1. DE (3)
2. AB (4)
3. GC (5)
4. CE (6)
5. AC (7)
6. DF (19)

edge number 5

AC

Answer

edge AC

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