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

What is the time complexity of building a binary heap from n unsorted elements with the bottom-up method?

  1. AO(n)
  2. BO(n log n)
  3. CO(log n)
  4. DO(n²)
3.Data structuresMedium

Why can a binary heap be stored in an array without any pointers?

  1. AAll keys in a heap are equal
  2. BHeaps never contain more than 100 elements
  3. CArrays automatically keep the keys in sorted order, so no pointers are needed
  4. DIt is a complete binary tree, so positions follow simple index formulas
4.Data structuresHard

A segment tree is typically used to:

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

A connected graph has 14 vertices and 19 edges. How many edges k must be removed to leave a spanning tree?

|V| = 14|E| = 19
k =
6.Graph theoryHard
2611572614ABCDEF

Use Dijkstra’s algorithm to find the length d(E) of the shortest path from A to E in the weighted graph.

start: Aend: E
d(E) =

3Solutions

1.Algorithms & complexity

Answer: C · 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

C · O(n)
2.Algorithms & complexity

Answer: A · O(n)

Why this is the answer

Summing sift-down costs over the levels gives a convergent series, so heapify runs in linear time.

Answer

A · O(n)
3.Data structures

Answer: D · It is a complete binary tree, so positions follow simple index formulas

Why this is the answer

Completeness means there are no gaps, so the children of index i are always at 2i + 1 and 2i + 2.

Answer

D · It is a complete binary tree, so positions follow simple index formulas
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 = 6

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 = 19 − 13 = 6

Answer

k = 6
6.Graph theory

Answer: d(E) = 10

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: 2, C: 6
B (2): C: 3, D: 17, E: 16
C (3): E: 10
E (10): D: 12, F: 16
D (12)
F (16)

Shortest path

A → B → C → E
d(E) = 10

Answer

d(E) = 10

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