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 running time of LSD radix sort on n integers with d digits in base k?

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

What is the running time of Dijkstra's algorithm with a binary heap on a graph with V vertices and E edges?

  1. AO((V + E) log V)
  2. BO(V³)
  3. CO(V + E)
  4. DO(V × E)
3.Data structuresMedium

How is a new element inserted into a binary heap?

  1. ARebuild the whole heap from scratch
  2. BPlace it at the root and sift it down
  3. CPlace it at the end of the array and sift it up while it violates the heap property
  4. DInsert it at its sorted position by shifting all of the larger elements one place right
4.Data structuresHard

For a hash table to work correctly, which rule must hash and equality follow?

  1. AEqual keys must have equal hash values
  2. BHash values must be prime numbers
  3. CDifferent keys must have different hash values
  4. DEqual hash values imply equal keys
5.Graph theoryMedium

How many different Hamilton circuits H does the complete graph K8 have? Circuits that differ only in the starting vertex or the direction count as the same.

K8
H =
6.Graph theoryHard
11510151121433ABCDEFG

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: B · O(d(n + k))

Why this is the answer

Each of the d passes is a counting sort costing O(n + k), so for fixed d and k the total is linear in n.

Answer

B · O(d(n + k))
2.Algorithms & complexity

Answer: A · O((V + E) log V)

Why this is the answer

There are V extract-min and up to E decrease-key operations, each O(log V); a Fibonacci heap improves this to O(E + V log V).

Answer

A · O((V + E) log V)
3.Data structures

Answer: C · Place it at the end of the array and sift it up while it violates the heap property

Why this is the answer

Adding at the end keeps the tree complete; swapping with the parent restores the order in O(log n) steps.

Answer

C · Place it at the end of the array and sift it up while it violates the heap property
4.Data structures

Answer: A · Equal keys must have equal hash values

Why this is the answer

If two equal keys hashed differently they would be stored in different buckets and lookups would fail; unequal keys may still share a hash.

Answer

A · Equal keys must have equal hash values
5.Graph theory

Answer: H = 2520

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.

Count

H = (8 − 1)!/2 = 5040/2 = 2520

Answer

H = 2520
6.Graph theory

Answer: d(E) = 22

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: 11, C: 5
C (5): G: 19
B (11)
G (19): D: 22, E: 22
D (22)
E (22): F: 23
F (23)

Shortest path

A → C → G → E
d(E) = 22

Answer

d(E) = 22

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