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 computing the n-th Fibonacci number with a bottom-up table?

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

Which algorithm finds a minimum spanning tree by repeatedly adding the cheapest edge that does not form a cycle?

  1. ADijkstra's algorithm
  2. BPrim's algorithm starting from one node
  3. CKruskal's algorithm
  4. DFloyd–Warshall
3.Data structuresMedium

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

  1. AFollow right children from the root until there is no right child
  2. BFollow left children from the root until there is no left child
  3. CVisit every leaf and compare them
  4. DLook at the root only
4.Data structuresHard

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

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

A tree has 4 vertices of degree 3, 3 vertices of degree 4, 3 vertices of degree 5, 2 vertices of degree 2, and all its other vertices are leaves (degree 1). How many leaves L does it have?

4 × deg 33 × deg 43 × deg 52 × deg 2
L =
6.Graph theoryHard
216104313177ABCDEF

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

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

3Solutions

1.Algorithms & complexity

Answer: D · O(n)

Why this is the answer

Filling the table from 0 to n takes one addition per entry, so the work is linear instead of exponential.

Answer

D · O(n)
2.Algorithms & complexity

Answer: C · Kruskal's algorithm

Why this is the answer

Kruskal sorts edges by weight and adds them if they connect different components; Prim grows one tree from a start vertex.

Answer

C · Kruskal's algorithm
3.Data structures

Answer: B · 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

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

Answer: D · 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

D · Move one pointer by one step and another by two steps and check whether they meet
5.Graph theory

Answer: L = 21

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.

Handshake lemma for a tree

L + 43 = 2(L + 12) − 2
L = 21

Answer

L = 21

Shortcut: L = 2 + Σ (d − 2) over the vertices with degree d ≥ 3.

6.Graph theory

Answer: W = 29

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

AB (2): take
DE (3): take
CE (4): take
BE (7): take
BD (10): skip, closes a cycle
DF (13): take

Total weight

W = 2 + 3 + 4 + 7 + 13 = 29

Answer

W = 29

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