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 is the time complexity of: for i = 1 to n: for j = 1 to i: do O(1) work?

  1. AO(n)
  2. BO(n²)
  3. CO(n³)
  4. DO(n log n)
2.Algorithms & complexityHard

For which knapsack variant does the greedy rule "take items in order of value per weight" always give an optimal answer?

  1. AThe 0/1 knapsack, where items must always be taken whole
  2. BBoth variants
  3. CThe fractional knapsack, where items may be split
  4. DNeither variant
3.Data structuresMedium

How do you find the minimum key in a binary search tree?

  1. AVisit every leaf and compare them
  2. BLook at the root only
  3. CFollow left children from the root until there is no left child
  4. DFollow right children from the root until there is no right child
4.Data structuresHard

A Fenwick tree (binary indexed tree) over an array of n numbers supports which operations in O(log n)?

  1. AInserting elements at any position
  2. BPrefix-sum queries and single-element updates
  3. CFinding the median in O(1)
  4. DSorting the array and binary search
5.Graph theoryMedium

A connected graph has 13 vertices and 18 edges. How many edges k must be removed to leave a spanning tree?

|V| = 13|E| = 18
k =
6.Graph theoryHard
811391214915ABCDEF

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(n²)

Why this is the answer

The inner body runs n(n + 1) / 2 times in total, which is still quadratic.

Answer

B · O(n²)
2.Algorithms & complexity

Answer: C · The fractional knapsack, where items may be split

Why this is the answer

When items can be split, the best ratio first is optimal; in 0/1 knapsack a high-ratio item can block a better combination, so dynamic programming is needed.

Answer

C · The fractional knapsack, where items may be split
3.Data structures

Answer: C · Follow left children from the root until there is no left child

Why this is the answer

The leftmost node has no smaller key in the tree, so it is the minimum; the cost is proportional to the height.

Answer

C · Follow left children from the root until there is no left child
4.Data structures

Answer: B · Prefix-sum queries and single-element updates

Why this is the answer

A Fenwick tree stores partial sums over ranges determined by the low bits of each index, so both prefix sums and point updates take O(log n).

Answer

B · Prefix-sum queries and single-element updates
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

13 − 1 = 12

Edges to remove

k = 18 − 12 = 6

Answer

k = 6
6.Graph theory

Answer: d(F) = 26

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: 8, C: 11
B (8): D: 17, E: 23
C (11)
D (17): F: 26
E (23)
F (26)

Shortest path

A → B → D → F
d(F) = 26

Answer

d(F) = 26

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