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 does the recurrence T(n) = T(n − 1) + O(1) solve to?

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

What is the difference between an undecidable problem and an intractable one?

  1. AAn undecidable problem has no algorithm for all inputs; an intractable one has one, but it is too slow
  2. BIntractable problems cannot be solved by any algorithm, while undecidable ones only need exponential time
  3. CUndecidable problems are in NP, while intractable problems are in P
  4. DBoth mean the problem has no algorithm at all
3.Data structuresMedium

Which collision resolution method stores colliding keys in a list at the same bucket?

  1. ADouble hashing
  2. BSeparate chaining
  3. CQuadratic probing
  4. DLinear probing
4.Data structuresHard

Deleting a node with two children from a binary search tree is usually done by:

  1. ARebuilding the subtree below it from scratch and reinserting every key it contained
  2. BSwapping it with the root
  3. CReplacing it with its in-order successor (or predecessor) and deleting that node
  4. DRemoving the whole subtree
5.Graph theoryMedium

A tree has 2 vertices of degree 3, 2 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?

2 × deg 32 × deg 52 × deg 2
L =
6.Graph theoryHard
2112363125ABCDEF

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: D · O(n)

Why this is the answer

There are n levels with constant work each, as in a recursive sum of an array.

Answer

D · O(n)
2.Algorithms & complexity

Answer: A · An undecidable problem has no algorithm for all inputs; an intractable one has one, but it is too slow

Why this is the answer

The halting problem has no algorithm at all; an NP-hard problem such as TSP can be solved by brute force, just not in known polynomial time.

Answer

A · An undecidable problem has no algorithm for all inputs; an intractable one has one, but it is too slow
3.Data structures

Answer: B · Separate chaining

Why this is the answer

Chaining keeps a linked list (or other structure) per bucket; open addressing looks for another free slot in the table itself.

Answer

B · Separate chaining
4.Data structures

Answer: C · Replacing it with its in-order successor (or predecessor) and deleting that node

Why this is the answer

The in-order successor has at most one child, so it can be removed easily after copying its key into the node being deleted.

Answer

C · Replacing it with its in-order successor (or predecessor) and deleting that node
5.Graph theory

Answer: L = 10

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 + 20 = 2(L + 6) − 2
L = 10

Answer

L = 10

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

6.Graph theory

Answer: d(F) = 8

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: 2, C: 1
C (1): E: 7
B (2): D: 5
D (5): F: 8
E (7)
F (8)

Shortest path

A → B → D → F
d(F) = 8

Answer

d(F) = 8

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