Version 5.0 The LEDA User Manual

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int int real roots(const Polynomial& P, growing array& roots, algorithm type algorithm = isolating algorithm, bool is squarefree = true) same as above. real roots(const int Polynomial& iP, list& roots, algorithm type algorithm = isolating algorithm, bool is squarefree = true) returns all real roots of the polynomial iP. real diamond(int j, const Polynomial& P, algorithm type algorithm = isolating algorithm, bool is squarefree = true) returns the j-th smallest real root of the polynomial P . real diamond(rational l, rational u, const Polynomial& P, algorithm type algorithm = isolating algorithm, bool is squarefree = true) returns the real root of the polynomial P which is in the isolating interval [l,u]. real diamond short(rational l, rational u, const Polynomial& P, algorithm type algorithm = isolating algorithm, bool is squarefree = true) returns the real root of the polynomial P which is in the isolating interval [l,u]. Precondition: (u − l) < 1/4 real real diamond(int j, const int Polynomial& iP, algorithm type algorithm = isolating algorithm, bool is squarefree = true) returns the j-th smallest real root of the polynomial iP. diamond(rational l, rational u, const int Polynomial& iP, algorithm type algorithm = isolating algorithm, bool is squarefree = true) returns the real root of the polynomial iP which is in the isolating interval [l,u]. real abs(const real& x) absolute value of x real sqr(const real& x) square of x real dist(const real& x, const real& y) euclidean distance of point (x,y) to the origin real powi(const real& x, int n) x n , i.e., n.th power of x
integer floor(const real& x) integer ceil(const real& x) returns the largest integer smaller than or equal to x. returns the smallest integer greater than or equal to x. rational small rational between(const real& x, const real& y) returns a rational number between x and y with the smallest available denominator. Note that the denominator does not need to be strictly minimal over all possible rationals. rational small rational near(const real& x, double eps) 5. Implementation returns small rational between(x − eps, x + eps). A real is represented by the expression which defines it and an interval inclusion I that contains the exact value of the real. The arithmetic operators +, −, ∗, d √ take constant time. When the sign of a real number needs to be determined, the data type first computes a number q, if not already given as an argument to sign, such that |x| ≤ 2 −q implies x = 0. The bound q is computed as described in [79]. Using bigfloat arithmetic, the data type then computes an interval I of maximal length 2 −q that contains x. If I contains zero, then x itself is equal to zero. Otherwise, the sign of any point in I is returned as the sign of x. Two shortcuts are used to speed up the computation of the sign. Firstly, if the initial interval approximation already suffices to determine the sign, then no bigfloat approximation is computed at all. Secondly, the bigfloat approximation is first computed only with small precision. The precision is then roughly doubled until either the sign can be decided (i.e., if the current approximation interval does not contain zero) or the full precision 2 −q is reached. This procedure makes the sign computation of a real number x adaptive in the sense that the running time of the sign computation depends on the complexity of x. 6. Example We give two typical examples for the use of the data type real that arise in Computational geometry. We admit that a certain knowledge about Computational geometry is required for their full understanding. The examples deal with the Voronoi diagram of line segments and the intersection of line segments, respectively. The following incircle test is used in the computation of Voronoi diagrams of line segments [17, 14]. For i, 1 ≤ i ≤ 3, let l i : a i x + b i y + c i = 0 be a line in two–dimensional space and let p = (0, 0) be the origin. In general, there are two circles passing through p and touching l 1 and l 2 . The centers of these circles have homogeneuos coordinates (x v , y v , z v ), where √ x v = a 1 c 2 + a 2 c 1 ± sign(s) 2c 1 c 2 ( √ N + D)
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Version 5.0 The LEDA User Manual Al
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4.14 Socket Streambuffer ( socket s
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9.11 Move-To-Front Coder II ( MTF2C
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13.1.2 Handles and Iterators . . .
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16.7 Simplices in 3D-Space ( d3 sim
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License Terms and Availability Any
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7. LEDA is available from Algorithm
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Chapter 2 Basics An extended versio
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XYZ y(x1, ... ,xt); ,tk > and uses
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2.3.1 Linear Orders Many data types
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dictionary D0; // default ordering
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2.3.3 Implementation Parameters Man
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lookup returned the location where
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list::iterator defines the iterator
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• internal (LEDA/incl/internal/)
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string s(const char ∗ p); string
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ool bool bool bool istream& ostream
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Page 42 and 43: andom source& S ≫ char& x random
Page 44 and 45: int R.set weight(int i, int g) sets
Page 46 and 47: 4.10 Memory Allocator ( leda alloca
Page 48 and 49: 4.11 Error Handling ( error ) LEDA
Page 50 and 51: 4.12 Files and Directories ( file )
Page 52 and 53: 4.13 Sockets ( leda socket ) 1. Def
Page 54 and 55: ool S.receive string(string& s) rec
Page 56 and 57: ool sb.failed( ) returns whether a
Page 58 and 59: 4.15 Some Useful Functions ( misc )
Page 60 and 61: 4.16 Timer ( timer ) 1. Definition
Page 62 and 63: unsigned fibonacci(unsigned n) { st
Page 64 and 65: unsigned fibonacci(unsigned n) { st
Page 66 and 67: int Hash(const two tuple& p) hash f
Page 68 and 69: 3. Creation four tuple p; creates a
Page 70 and 71: 4.21 A date interface ( date ) 1. D
Page 72 and 73: 3. Creation date D; creates an inst
Page 74 and 75: string D.get month name( ) returns
Page 76 and 77: Now we show an example in which dif
Page 78 and 79: integer integer a(const char ∗ s)
Page 80 and 81: 5.2 Rational Numbers ( rational ) 1
Page 82 and 83: 5.3 The data type bigfloat ( bigflo
Page 84 and 85: ounding modes bigfloat :: get round
Page 86 and 87: ostream& ostream& os ≪ const bigf
Page 88 and 89: double x.get double error( ) bigflo
Page 92 and 93: √ y v = b 1 c 2 + b 2 c 1 ± sign
Page 94 and 95: 5.5 Interval Arithmetic in LEDA ( i
Page 96 and 97: void x.set midpoint(VOLATILE I doub
Page 98 and 99: 5.7 The mod kernel of type residual
Page 100 and 101: int residual :: required primetable
Page 102 and 103: 5.9 A Floating Point Filter ( float
Page 104 and 105: 5.10 Double-Valued Vectors ( vector
Page 106 and 107: double v.zcoord( ) returns the seco
Page 108 and 109: matrix matrix matrix vector M + con
Page 110 and 111: 5.12 Vectors with Integer Entries (
Page 112 and 113: 5.13 Matrices with Integer Entries
Page 114 and 115: integer matrix inverse(const intege
Page 116 and 117: distinct representatives and linear
Page 118 and 119: at vector v(const array& A); introd
Page 120 and 121: int compare by angle(const rat vect
Page 122 and 123: 5.15 Real-Valued Vectors ( real vec
Page 124 and 125: int compare by angle(const real vec
Page 126 and 127: eal matrix M + const real matrix& M
Page 128 and 129: 5.17.2 Integration double integrate
Page 130 and 131: array A(int low, const E& x, const
Page 132 and 133: int A.binary locate(int (∗cmp)(co
Page 134 and 135: 6.3 Stacks ( stack ) 1. Definition
Page 136 and 137: 6.5 Bounded Stacks ( b stack ) 1. D
Page 138 and 139: 6.7 Linear Lists ( list ) 1. Defini
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const E& L.pop front( ) same as L.p
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Sorting and Searching void L.sort(i
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Input and Output void L.read(istrea
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6.8 Singly Linked Lists ( slist ) 1
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6.9 Sets ( set ) 1. Definition An i
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4. Implementation Sets are implemen
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int set S | const int set& T return
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d int set S − const d int set& T
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6.12 Partitions ( partition ) 1. De
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6.13 Parameterized Partitions ( Par
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void D.evert(vertex v) makes v the
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4. Implementation Dynamic collectio
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dictionary D(int (∗cmp)(const K&
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7.2 Dictionary Arrays ( d array ) 1
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{ d_array dic; dic["hello"] = "hall
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ool A.empty( ) returns true if A is
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Note that it is not possible to ite
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void M.clear( ) clears M by making
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p dictionary D.change inf(p dic ite
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void D.del(const K& k) deletes the
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3. Creation sortseq S; creates an i
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seq item S.max item( ) returns the
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Iteration forall items(it, S) { “
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Chapter 8 Priority Queues 8.1 Prior
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6. Example Dijkstra’s Algorithm (
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int Q.size( ) returns the size of Q
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Assume that you want to send the fi
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9.1 Adaptive Arithmetic Coder ( A0C
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uint32 C.get scale threshold( ) uin
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uint32 C.decode memory chunk(const
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3. Creation PPMIICoder C(streambuf
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9.4 Deflation/Inflation Coder ( Def
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void C.set compression level(int le
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void void C.set src stream(streambu
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streambuf ∗ C.get src stream( ) r
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streambuf ∗ C.get src stream( ) r
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uint32 C.decode memory chunk(const
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uint32 C.encode memory chunk(const
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9.10 Move-To-Front Coder ( MTFCoder
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9.11 Move-To-Front Coder II ( MTF2C
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9.12 RLE for Runs of Zero ( RLE0Cod
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9.13 Checksummers ( checksummer bas
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ool C.checksum is valid( ) returns
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9.19 Encoding Output Stream ( encod
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ostream& os.seekp(streamoff off , i
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9.23 Coder Pipes ( CoderPipe2 ) 1.
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void C.set coder1(Coder1 ∗ c1 , b
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void void C.set src stream(streambu
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3. Operations Standard Operations v
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string out_file1 = tmp_file_name(),
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void mb.truncate(streamsize n) also
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e able to change your message witho
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Then you can use the CryptAutoDecod
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10.1 Secure Byte String ( CryptByte
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CryptKey k(const byte ∗ bytes, ui
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10.3 Encryption and Decryption with
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10.4 Example for a Stream-Cipher (
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uint16 void C.get accepted key size
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3. Operations Standard Operations v
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10.6 Automatic Decoder supporting C
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void C.add keys in file(const char
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secure socket streambuf sb(leda soc
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Chapter 11 Graphs and Related Data
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2. Creation graph G; creates an obj
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edge G.first in edge(node v) return
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void G.hide node(node v) removes no
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void G.bucket sort nodes(int l, int
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edge G.new map edge(edge e1 , edge
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edge edge edge G.new edge(node v, e
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void void G.print(string s, ostream
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graph algorithms, i.e., algorithms
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void void G.sort nodes(const list&
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• Opposite Graphs (opposite graph
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4. Operations The interface consist
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int main () { static_graph G; array
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3. Implementation see section 11.2.
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list M.triangulate( ) triangulates
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void M.assign(face f, const ftype&
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void A.init(const graph t& G, int n
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ool M.use node data(const graph t&
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ool M.use edge data(const graph t&
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E& M[face f] returns the variable M
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4. Implementation Node matrices for
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4. Implementation Node maps are imp
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11.17 Sets of Edges ( edge set ) 1.
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node L.cyclic pred(node v) returns
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11.20 Node Priority Queues ( node p
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11.21 Bounded Node Priority Queues
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11.22 Graph Generators ( graph gen
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uedges. For n > 3, a random maximal
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void random planar graph(graph& G,
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ool Is Bidirected(const graph& G) r
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list Delete Loops(graph& G) returns
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11.25 Dynamic Markov Chains ( dynam
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has a source and a target. These ar
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void parser.append(const char ∗ k
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11.27 The LEDA graph input/output f
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arithmetic demand for each function
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GRAPH TRANSITIVE CLOSURE(const gra
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template bool SHORTEST PATH T(cons
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template bool ALL PAIRS SHORTEST P
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The algorithms have the following a
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inline NT MAX FLOW GRH T(const grap
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12.5 Minimum Cut ( min cut ) A cut
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that the bipartition (A, B) is give
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The pre-instantiations for number t
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template bool CHECK MIN WEIGHT ASS
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in general graph. You may skip the
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Worst-Case Running Time: All functi
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list MIN WEIGHT PERFECT MATCHING T(
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12.10 Stable Matching ( stable matc
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12.11 Minimum Spanning Trees ( min
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12.13 Algorithms for Planar Graphs
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ool bool Is CCW Ordered(const graph
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ool void void void TUTTE EMBEDDING(
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Chapter 13 Graphs and Iterators 13.
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The purpose of each iterator is the
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template void set_and_check (graph
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In addition to the algorithm mentio
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13.3 Edge Iterators ( EdgeIt ) 1. D
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#include < LEDA/graph/graph iterato
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OutAdjIt OutAdjIt OutAdjIt it(const
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OutAdjIt& −−it performs one ste
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ool it.eol( ) returns !it.valid( )
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void it.del( ) deletes the marked l
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3. Operations void fc.init(const le
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We would have to write something li
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Equal is a class that compares two
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used only as a “Trojan horse,”
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typename DataAccessor :: value type
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13.14 Constant Accessors ( constant
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13.16 Node Attribute Accessors ( no
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GIT BFS< OutAdjIt, Queuetype, Mark
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3. dfs grow depth: a new adjacency
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• indegree stores for every node
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• Mark is a data accessor that ha
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OutAdjIt algorithm.curr adj( ) retu
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14.1 Points ( point ) 1. Definition
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point p.reflect(const point& q, con
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ool contained in simplex(const arra
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4. Operations point s.start( ) retu
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segment s.rotate90(const point& q,
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double r.direction( ) returns the d
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14.4 Straight Lines ( line ) 1. Def
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line l.translate(const vector& v) r
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14.5 Circles ( circle ) 1. Definiti
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circle C + const vector& v returns
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14.6 Polygons ( POLYGON ) 1. Defini
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const list& P.segments( ) returns t
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ool P.inside(const POINT & p) bool
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14.7 Generalized Polygons ( GEN POL
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GEN POLYGON GEN POLYGON :: make wea
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int P.side of(const POINT & p) regi
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14.8 Triangles ( triangle ) 1. Defi
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triangle t.rotate90(const point& q,
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double r.height( ) returns the heig
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14.10 Rational Points ( rat point )
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at point p.reflect(const rat point&
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int side of halfspace(const rat poi
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at segment s(const segment& s1 , in
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ational s.y abs( ) returns the y-ab
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14.12 Rational Rays ( rat ray ) 1.
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ool r.contains(const rat segment& s
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void l.normalize( ) simplifies the
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at line p bisector(const rat point&
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at circle C(const circle& c, int pr
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14.15 Rational Triangles ( rat tria
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at triangle t − const rat vector&
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list r.vertices( ) returns the vert
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list r.intersection(const rat recta
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int p.orientation(const real point&
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eal area(const real point& a, const
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14.18 Real Segments ( real segment
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ool s.intersection(const real segme
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14.19 Real Rays ( real ray ) 1. Def
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Non-Member Functions int orientatio
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4. Operations real point l.point1(
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int cmp slopes(const real line& l1
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eal circle C(const circle& c, int p
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ool radical axis(const real circle&
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eal point t.point3( ) returns the t
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14.23 Iso-oriented Real Rectangles
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eal rectangle r.translate(real dx,
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14.24 Geometry Algorithms ( geo alg
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edge TRIANGULATE PLANE MAP(GRAPH &
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gen polygon MINKOWSKI SUM(const pol
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ool MIN WIDTH ANNULUS(const list& L
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double CLOSEST PAIR(list& L, point&
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14.25 Transformation ( TRANSFORM )
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TRANSFORM reflection(const POINT& q
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void random points in unit cube(int
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void random points on unit circle(i
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const rat point& p.to rat point( )
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circle segment cs(const rat circle&
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list cs.intersection(const r circle
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14.29 Polygons with circular edges
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CHECK TYPE P.check simplicity( ) ch
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ool P.inside(const r circle point&
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circle gen polygon P (KIND k); crea
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list P.intersection(const rat line&
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double P.approximate area( ) approx
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const I& D.inf(dic2 item it) return
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15.2 Point Sets and Delaunay Triang
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ool T.is hull edge(edge e) as above
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list T.range search(node v, const P
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15.3 Sets of Intervals ( interval s
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15.4 Sets of Parallel Segments ( se
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15.5 Sets of Parallel Rational Segm
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15.6 Planar Subdivisions ( subdivis
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16.1 Points in 3D-Space ( d3 point
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d3 point midpoint(const d3 point& a
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ool outside sphere(const d3 point&
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ool bool r.project xz(ray& m) if th
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segment s.project xy( ) returns the
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ool l.project(const d3 point& p, co
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double p.sqr dist(const d3 point& q
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16.6 Spheres in 3D-Space ( d3 spher
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16.7 Simplices in 3D-Space ( d3 sim
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16.8 Rational Points in 3D-Space (
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d3 rat point p.translate(const rat
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ool coplanar(const d3 rat point& a,
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d3 rat point random d3 rat point in
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void random d3 rat points on segmen
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ool r.project xy(rat ray& m) bool r
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ool l.project yz(rat line& m) if th
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16.11 Rational Segments in 3D-Space
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d3 rat segment s + const rat vector
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d3 plane p.to float( ) returns a fl
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16.13 Rational Spheres ( d3 rat sph
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16.14 Rational Simplices ( d3 rat s
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16.15 3D Convex Hull Algorithms ( d
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Chapter 17 Graphics This section de
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17.2 Windows ( window ) 1. Definiti
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18. The buttons per line parameter
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void W.display(window& W 0 , int x,
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void W.set show coord object(const
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int W.menu bar height( ) returns th
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void W.draw ray(point p, point q, l
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void W.draw filled ellipse(double x
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void void void void void W.draw rou
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void void void W.draw edge(double x
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void W.screenshot(string fname, boo
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int W.read mouse seg(const point& p
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int W.read event(int& val, double&
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window& W ≫ segment& s reads a se
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panel item W.bool item(string s, bo
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panel item W.int item(string s, int
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int W.button(string s, const char
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int W.button(string s, window& M, c
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void void void W.redraw panel(panel
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17.4 Menues ( menu ) 1. Definition
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17.5 Postscript Files ( ps file ) 1
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shape: the shape of the node (type
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void gw.display(int x, int y) displ
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param type gw.set param(node v, par
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string gw.set edge index format(str
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void void gw.transform layout(node
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int bool bool gw.load layout(istrea
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void void gw.set edge slider handle
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int bool bool bool bool bool bool g
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void gw.restore all attributes( ) .
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6. edge index format format This li
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of GraphWin) are 0 (central pos) 1
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17.7.1 A complete example LEDA.GRAP
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17.8 Geometry Windows ( GeoWin ) 1.
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0 will be drawn on top in the order
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geowin new scene and geowin get obj
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e inserted in the result scene. In
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geo_scene sc_points = gw.new_scene(
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} }; int main() { GeoWin gw; list L
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virtual bool is_running(const GeoWi
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geo scene void int GW.get scene wit
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color GW.get fill color(geo scene s
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void∗ GW.set client data(geo scen
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color GW.set obj color(GeoBaseScene
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template string GW.set obj label(G
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string string GW.get bg pixmap( ) r
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• A DRAG (button not released)
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template bool GW.set post del hand
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geowin_gui_rat_transform geowin_gui
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ool GW.del(geo scenegroup GS , geo
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void GW.set d3 fcn(geo scene sc, vo
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17.9 Windows for 3d visualization (
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int D.move( ) animates the contents
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Chapter 18 Implementations 18.1 Lis
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void clear(); int size() const; };
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18.2.3 Sorted Sequences Any class s
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libGeoW is the LEDA library providi
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(b) Type VCVARS32 (VSVARS32). 2. Go
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LEDA MEMORY STD: If this is set, th
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When using graphics on Solaris syst
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(7) Double click on prog.cpp If you
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• extend LIB by Otherwise add ne
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and at least one of the following d
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• Copy leda .dll to the bin\ subd
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Preparations To install LEDA you on
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• libg .lib libl .lib for program
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cl -TP prog.c Programs using grap
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Automatic Setting Include header fi
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The default value is ”/MLd”, al
Page 752 and 753:
2. Windows 95/98: (a) Add the line
Page 754 and 755:
Preparations To install LEDA you on
Page 756 and 757:
A.16 Platforms When this manual was
Page 758 and 759:
(b) For objects x and y of an indep
Page 760 and 761:
A and B are different objects: A ==
Page 762 and 763:
if (p.y > q.y) return 1; return 0;
Page 764 and 765:
partypes={no, yes} Determines how p
Page 766 and 767:
Stacks ( stack ) 1. Definition An i
Page 769 and 770:
Bibliography [1] H. Alt, N. Blum, K
Page 771 and 772:
[31] I. Fary: “On Straight Line R
Page 773 and 774:
[67] M. Mignotte: Mathematics for C
Page 775 and 776:
Index Symbols L A TEX . . . . . . .
Page 777 and 778:
adj nodes(...) graph . . . . . . .
Page 779 and 780:
circle gen polygon . . . . . . . .
Page 781 and 782:
at ray . . . . . . . . . . . . . .
Page 783 and 784:
CBCCoder< BlkCipher > . . . . . . .
Page 785 and 786:
disconnect() leda socket . . . . .
Page 787 and 788:
edges() GEN POLYGON . . . . . . . .
Page 789 and 790:
finger locate pre...(...) sortseq .
Page 791 and 792:
checksummer base . . . . . . . . .
Page 793 and 794:
OMACCoder< BlkCipher > . . . . . .
Page 795 and 796:
get stack() GIT DFS . . . . . . . .
Page 797 and 798:
in pred(...) graph . . . . . . . .
Page 799 and 800:
eal triangle . . . . . . . . . . .
Page 801 and 802:
at segment . . . . . . . . . . . .
Page 803 and 804:
eal rectangle . . . . . . . . . . .
Page 805 and 806:
mw matching . . . . . . . . . . . .
Page 807 and 808:
outdeg(...) graph . . . . . . . . .
Page 809 and 810:
pos(...) POINT SET . . . . . . . .
Page 811 and 812:
eal matrix . . . . . . . . . . . .
Page 813 and 814:
FaceIt . . . . . . . . . . . . . .
Page 815 and 816:
save gw(...) GraphWin . . . . . . .
Page 817 and 818:
set edge index fo...(...) GraphWin
Page 819 and 820:
GeoWin . . . . . . . . . . . . . .
Page 821 and 822:
DeflateCoder . . . . . . . . . . .
Page 823 and 824:
planar map . . . . . . . . . . . .
Page 825 and 826:
d3 rat point . . . . . . . . . . .
Page 827 and 828:
GraphWin . . . . . . . . . . . . .
Page 829 and 830:
at segment . . . . . . . . . . . .
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Version 5.0 The LEDA User Manual

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