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

Breadth-first search on an unweighted graph finds:

  1. AShortest paths in graphs that contain negative edge weights
  2. BShortest paths in number of edges from the start node
  3. CThe minimum spanning tree
  4. DA topological order in all graphs
2.Algorithms & complexityHard

Taking both endpoints of every edge in a maximal matching gives a vertex cover. How good is this cover?

  1. AOf unknown quality compared to the optimum
  2. BAt most twice the size of a minimum vertex cover
  3. CAt most log n times the minimum
  4. DAlways a minimum vertex cover
3.Data structuresMedium

In a hash table using open addressing, what is the maximum possible load factor?

  1. A1
  2. B2
  3. C0.5
  4. DThere is no maximum
4.Data structuresHard

How can you detect whether a singly linked list contains a cycle using O(1) extra memory?

  1. AMove one pointer by one step and another by two steps and check whether they meet
  2. BSort the list and look for duplicates
  3. CStore every visited node in a hash set
  4. DCount the nodes until the count exceeds the list length field
5.Graph theoryMedium

A connected graph has 16 vertices and 21 edges. How many edges k must be removed to leave a spanning tree?

|V| = 16|E| = 21
k =
6.Graph theoryHard
837141019118ABCDEF

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

|V| = 6|E| = 8
W =

3Solutions

1.Algorithms & complexity

Answer: B · Shortest paths in number of edges from the start node

Why this is the answer

BFS visits nodes in order of distance (edges) from the source, so it finds shortest unweighted paths in O(V + E).

Answer

B · Shortest paths in number of edges from the start node
2.Algorithms & complexity

Answer: B · At most twice the size of a minimum vertex cover

Why this is the answer

The matching edges share no vertices, so the optimum needs at least one vertex per edge, while this method takes two: a 2-approximation.

Answer

B · At most twice the size of a minimum vertex cover
3.Data structures

Answer: A · 1

Why this is the answer

With open addressing all keys are stored in the table itself, so n cannot exceed m; separate chaining can have a load factor above 1.

Answer

A · 1
4.Data structures

Answer: A · Move one pointer by one step and another by two steps and check whether they meet

Why this is the answer

In Floyd's algorithm, if there is a cycle the fast pointer eventually catches up with the slow one; otherwise the fast pointer reaches null.

Answer

A · Move one pointer by one step and another by two steps and check whether they meet
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

16 − 1 = 15

Edges to remove

k = 21 − 15 = 6

Answer

k = 6
6.Graph theory

Answer: W = 35

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

EF (1): take
AC (3): take
BC (7): take
AB (8): skip, closes a cycle
DE (10): take
CE (14): take

Total weight

W = 1 + 3 + 7 + 10 + 14 = 35

Answer

W = 35

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