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

In a sorted array, you look for two elements that sum to a target by moving one pointer from each end inwards. What is the time complexity?

  1. AO(log n)
  2. BO(n)
  3. CO(n log n)
  4. DO(n²)
2.Algorithms & complexityHard

How do n! and 2ⁿ compare for large n?

  1. An! is O(n²)
  2. BThey grow at the same rate
  3. C2ⁿ grows faster than n!
  4. Dn! grows faster than 2ⁿ
3.Data structuresMedium

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

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

A segment tree is typically used to:

  1. AStore a graph's edges
  2. BSort strings alphabetically by storing each prefix in a separate tree node
  3. CAnswer range queries (such as range sums) and updates in O(log n)
  4. DImplement a stack
5.Graph theoryMedium

A connected graph has 7 vertices and 18 edges. How many edges k must be removed to leave a spanning tree?

|V| = 7|E| = 18
k =
6.Graph theoryHard
1214510937618ABCDEFG

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 · O(n)

Why this is the answer

If the sum is too small the left pointer moves right, if too large the right pointer moves left, so there are at most n steps.

Answer

B · O(n)
2.Algorithms & complexity

Answer: D · n! grows faster than 2ⁿ

Why this is the answer

For n ≥ 4, n! exceeds 2ⁿ, and the ratio grows without bound because each extra factor of n! is larger than 2.

Answer

D · n! grows faster than 2ⁿ
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: C · Answer range queries (such as range sums) and updates in O(log n)

Why this is the answer

A segment tree stores aggregates for intervals, supporting range queries and point updates in O(log n) time.

Answer

C · Answer range queries (such as range sums) and updates in O(log n)
5.Graph theory

Answer: k = 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.

Edges of a spanning tree

7 − 1 = 6

Edges to remove

k = 18 − 6 = 12

Answer

k = 12
6.Graph theory

Answer: d(F) = 9

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: 1, C: 2
B (1): D: 6
C (2): E: 12, G: 8
D (6): F: 9, G: 7
G (7)
F (9)
E (12)

Shortest path

A → B → D → F
d(F) = 9

Answer

d(F) = 9

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