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

With coins of 1, 3 and 4 and an amount of 6, how many coins does the greedy "largest coin first" method use, compared to the optimum?

  1. AGreedy uses 3 coins; the optimum is 2
  2. BGreedy uses 4 coins; the optimum is 3
  3. CGreedy uses 2 coins; the optimum is 2
  4. DGreedy uses 3 coins; the optimum is 3
2.Algorithms & complexityHard

Karatsuba multiplication satisfies T(n) = 3T(n/2) + O(n). What does this solve to?

  1. AΘ(nlog₂ 3), about n1.585
  2. BΘ(n²)
  3. CΘ(n3)
  4. DΘ(n log n)
3.Data structuresMedium

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

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

Which statement about heapsort is correct?

  1. AIt needs a binary search tree as extra storage, so it uses O(n) extra memory
  2. BIt is stable and needs O(n) extra memory
  3. CIt runs in O(n) time on any input
  4. DIt sorts in place in O(n log n) worst-case time but is not stable
5.Graph theoryMedium

A connected graph has 14 vertices and 22 edges. How many edges k must be removed to leave a spanning tree?

|V| = 14|E| = 22
k =
6.Graph theoryHard
9312131511121172ABCDEFG

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: A · Greedy uses 3 coins; the optimum is 2

Why this is the answer

Greedy takes 4 + 1 + 1 = 3 coins, but 3 + 3 uses only 2, so greedy is not optimal for this coin system; dynamic programming is.

Answer

A · Greedy uses 3 coins; the optimum is 2
2.Algorithms & complexity

Answer: A · Θ(nlog₂ 3), about n1.585

Why this is the answer

Since log₂ 3 ≈ 1.585 > 1, the leaves dominate by the master theorem, beating the O(n²) schoolbook method.

Answer

A · Θ(nlog₂ 3), about n1.585
3.Data structures

Answer: D · 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

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

Answer: D · It sorts in place in O(n log n) worst-case time but is not stable

Why this is the answer

Heapsort builds a max-heap and extracts the maximum n times within the same array; swaps over long distances can reorder equal keys.

Answer

D · It sorts in place in O(n log n) worst-case time but is not stable
5.Graph theory

Answer: k = 9

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

14 − 1 = 13

Edges to remove

k = 22 − 13 = 9

Answer

k = 9
6.Graph theory

Answer: d(F) = 24

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: 9, C: 3
C (3): G: 10
B (9): D: 22
G (10): E: 12
E (12): F: 24
D (22)
F (24)

Shortest path

A → C → G → E → F
d(F) = 24

Answer

d(F) = 24

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