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

How does Prim's algorithm build a minimum spanning tree?

  1. AIt removes the most expensive edge until no cycles remain
  2. BIt sorts all edges by weight and adds each one unless it would form a cycle in the forest
  3. CIt grows one tree from a start vertex, adding the cheapest edge that leaves the tree
  4. DIt runs breadth-first search and keeps the BFS tree
2.Algorithms & complexityHard

Topological sorting is possible only for:

  1. ADirected acyclic graphs
  2. BUndirected graphs
  3. CComplete graphs
  4. DGraphs with negative weights
3.Data structuresMedium

What happens when a hash table rehashes after its load factor exceeds a threshold?

  1. AThe oldest keys are deleted to make room
  2. BA larger table is allocated and every key is inserted again
  3. CThe existing keys are sorted in place
  4. DOnly the colliding keys are moved to a new overflow list at the end of the table
4.Data structuresHard

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

  1. AHash values must be prime numbers
  2. BDifferent keys must have different hash values
  3. CEqual keys must have equal hash values
  4. DEqual hash values imply equal keys
5.Graph theoryMedium

How many different Hamilton circuits H does the complete graph K6 have? Circuits that differ only in the starting vertex or the direction count as the same.

K6
H =
6.Graph theoryHard
14594201712101815ABCDEFG

Find the total weight W of a minimum spanning tree of the weighted graph.

|V| = 7|E| = 10
W =

3Solutions

1.Algorithms & complexity

Answer: C · It grows one tree from a start vertex, adding the cheapest edge that leaves the tree

Why this is the answer

Prim's algorithm, with a priority queue of crossing edges, runs in O(E log V) with a binary heap.

Answer

C · It grows one tree from a start vertex, adding the cheapest edge that leaves the tree
2.Algorithms & complexity

Answer: A · Directed acyclic graphs

Why this is the answer

A topological order exists if and only if the directed graph has no cycles; it can be found in O(V + E) with DFS or Kahn's algorithm.

Answer

A · Directed acyclic graphs
3.Data structures

Answer: B · A larger table is allocated and every key is inserted again

Why this is the answer

Since the index is computed modulo the table size, all keys must be rehashed into the new, larger table; with doubling this stays O(1) amortised.

Answer

B · A larger table is allocated and every key is inserted again
4.Data structures

Answer: C · 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

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

Answer: H = 60

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 = (6 − 1)!/2 = 120/2 = 60

Answer

H = 60
6.Graph theory

Answer: W = 55

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.

Kruskal’s algorithm

BD (4): take
AC (5): take
BC (9): take
EF (10): take
DF (12): take
AB (14): skip, closes a cycle
GD (15): take

Total weight

W = 4 + 5 + 9 + 10 + 12 + 15 = 55

Answer

W = 55

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