EMD

/*
    emd.c

    Last update: 3/14/98

    An implementation of the Earth Movers Distance.
    Based of the solution for the Transportation problem as described in
    "Introduction to Mathematical Programming" by F. S. Hillier and 
    G. J. Lieberman, McGraw-Hill, 1990.

    Copyright (C) 1998 Yossi Rubner
    Computer Science Department, Stanford University
    E-Mail: rubner@cs.stanford.edu   URL: http://vision.stanford.edu/~rubner
*/

#include <stdio.h>
#include <stdlib.h>
#include <math.h>

#include "emd.h"

#define DEBUG_LEVEL 0
/*
 DEBUG_LEVEL:
   0 = NO MESSAGES
   1 = PRINT THE NUMBER OF ITERATIONS AND THE FINAL RESULT
   2 = PRINT THE RESULT AFTER EVERY ITERATION
   3 = PRINT ALSO THE FLOW AFTER EVERY ITERATION
   4 = PRINT A LOT OF INFORMATION (PROBABLY USEFUL ONLY FOR THE AUTHOR)
*/


#define MAX_SIG_SIZE1  (MAX_SIG_SIZE+1)  /* FOR THE POSIBLE DUMMY FEATURE */

/* NEW TYPES DEFINITION */

/* node1_t IS USED FOR SINGLE-LINKED LISTS */
typedef struct node1_t {
  int i;
  double val;
  struct node1_t *Next;
} node1_t;

/* node1_t IS USED FOR DOUBLE-LINKED LISTS */
typedef struct node2_t {
  int i, j;
  double val;
  struct node2_t *NextC;               /* NEXT COLUMN */
  struct node2_t *NextR;               /* NEXT ROW */
} node2_t;



/* GLOBAL VARIABLE DECLARATION */
static int _n1, _n2;                          /* SIGNATURES SIZES */
static float _C[MAX_SIG_SIZE1][MAX_SIG_SIZE1];/* THE COST MATRIX */
static node2_t _X[MAX_SIG_SIZE1*2];            /* THE BASIC VARIABLES VECTOR */
/* VARIABLES TO HANDLE _X EFFICIENTLY */
static node2_t *_EndX, *_EnterX;
static char _IsX[MAX_SIG_SIZE1][MAX_SIG_SIZE1];
static node2_t *_RowsX[MAX_SIG_SIZE1], *_ColsX[MAX_SIG_SIZE1];
static double _maxW;
static float _maxC;

/* DECLARATION OF FUNCTIONS */
static float init(signature_t *Signature1, signature_t *Signature2,
		  float (*Dist)(feature_t *, feature_t *));
static void findBasicVariables(node1_t *U, node1_t *V);
static int isOptimal(node1_t *U, node1_t *V);
static int findLoop(node2_t **Loop);
static void newSol();
static void russel(double *S, double *D);
static void addBasicVariable(int minI, int minJ, double *S, double *D, 
			     node1_t *PrevUMinI, node1_t *PrevVMinJ,
			     node1_t *UHead);
#if DEBUG_LEVEL > 0
static void printSolution();
#endif


/******************************************************************************
float emd(signature_t *Signature1, signature_t *Signature2,
	  float (*Dist)(feature_t *, feature_t *), flow_t *Flow, int *FlowSize)
  
where

   Signature1, Signature2  Pointers to signatures that their distance we want
              to compute.
   Dist       Pointer to the ground distance. i.e. the function that computes
              the distance between two features.
   Flow       (Optional) Pointer to a vector of flow_t (defined in emd.h) 
              where the resulting flow will be stored. Flow must have n1+n2-1
              elements, where n1 and n2 are the sizes of the two signatures
              respectively.
              If NULL, the flow is not returned.
   FlowSize   (Optional) Pointer to an integer where the number of elements in
              Flow will be stored
              
******************************************************************************/

float emd(signature_t *Signature1, signature_t *Signature2,
	  float (*Dist)(feature_t *, feature_t *),
	  flow_t *Flow, int *FlowSize)
{
  int itr;
  double totalCost;
  float w;
  node2_t *XP;
  flow_t *FlowP;
  node1_t U[MAX_SIG_SIZE1], V[MAX_SIG_SIZE1];

  w = init(Signature1, Signature2, Dist);

#if DEBUG_LEVEL > 1
  printf("/nINITIAL SOLUTION:/n");
  printSolution();
#endif
 
  if (_n1 > 1 && _n2 > 1)  /* IF _n1 = 1 OR _n2 = 1 THEN WE ARE DONE */
    {
      for (itr = 1; itr < MAX_ITERATIONS; itr++)
	{
	  /* FIND BASIC VARIABLES */
	  findBasicVariables(U, V);
	  
	  /* CHECK FOR OPTIMALITY */
	  if (isOptimal(U, V))
	    break;
	  
	  /* IMPROVE SOLUTION */
	  newSol();
	  
#if DEBUG_LEVEL > 1
	  printf("/nITERATION # %d /n", itr);
	  printSolution();
#endif
	}

      if (itr == MAX_ITERATIONS)
	fprintf(stderr, "emd: Maximum number of iterations has been reached (%d)/n",
		MAX_ITERATIONS);
    }

  /* COMPUTE THE TOTAL FLOW */
  totalCost = 0;
  if (Flow != NULL)
    FlowP = Flow;
  for(XP=_X; XP < _EndX; XP++)
    {
      if (XP == _EnterX)  /* _EnterX IS THE EMPTY SLOT */
	continue;
      if (XP->i == Signature1->n || XP->j == Signature2->n)  /* DUMMY FEATURE */
	continue;
      
      if (XP->val == 0)  /* ZERO FLOW */
	continue;

      totalCost += (double)XP->val * _C[XP->i][XP->j];
      if (Flow != NULL)
	{
	  FlowP->from = XP->i;
	  FlowP->to = XP->j;
	  FlowP->amount = XP->val;
	  FlowP++;
	}
    }
  if (Flow != NULL)
    *FlowSize = FlowP-Flow;

#if DEBUG_LEVEL > 0
  printf("/n*** OPTIMAL SOLUTION (%d ITERATIONS): %f ***/n", itr, totalCost);
#endif

  /* RETURN THE NORMALIZED COST == EMD */
  return (float)(totalCost / w);
}





/**********************
   init
**********************/
static float init(signature_t *Signature1, signature_t *Signature2, 
		  float (*Dist)(feature_t *, feature_t *))
{
  int i, j;
  double sSum, dSum, diff;
  feature_t *P1, *P2;
  double S[MAX_SIG_SIZE1], D[MAX_SIG_SIZE1];
 
  _n1 = Signature1->n;
  _n2 = Signature2->n;

  if (_n1 > MAX_SIG_SIZE || _n2 > MAX_SIG_SIZE)
    {
      fprintf(stderr, "emd: Signature size is limited to %d/n", MAX_SIG_SIZE);
      exit(1);
    }
  
  /* COMPUTE THE DISTANCE MATRIX */
  _maxC = 0;
  for(i=0, P1=Signature1->Features; i < _n1; i++, P1++)
    for(j=0, P2=Signature2->Features; j < _n2; j++, P2++) 
      {
	_C[i][j] = Dist(P1, P2);
	if (_C[i][j] > _maxC)
	  _maxC = _C[i][j];
      }
	
  /* SUM UP THE SUPPLY AND DEMAND */
  sSum = 0.0;
  for(i=0; i < _n1; i++)
    {
      S[i] = Signature1->Weights[i];
      sSum += Signature1->Weights[i];
      _RowsX[i] = NULL;
    }
  dSum = 0.0;
  for(j=0; j < _n2; j++)
    {
      D[j] = Signature2->Weights[j];
      dSum += Signature2->Weights[j];
      _ColsX[j] = NULL;
    }

  /* IF SUPPLY DIFFERENT THAN THE DEMAND, ADD A ZERO-COST DUMMY CLUSTER */
  diff = sSum - dSum;
  if (fabs(diff) >= EPSILON * sSum)
    {
      if (diff < 0.0)
	{
	  for (j=0; j < _n2; j++)
	    _C[_n1][j] = 0;
	  S[_n1] = -diff;
	  _RowsX[_n1] = NULL;
	  _n1++;
	}
      else
	{
	  for (i=0; i < _n1; i++)
	    _C[i][_n2] = 0;
	  D[_n2] = diff;
	  _ColsX[_n2] = NULL;
	  _n2++;
	}
    }

  /* INITIALIZE THE BASIC VARIABLE STRUCTURES */
  for (i=0; i < _n1; i++)
    for (j=0; j < _n2; j++)
	_IsX[i][j] = 0;
  _EndX = _X;
   
  _maxW = sSum > dSum ? sSum : dSum;

  /* FIND INITIAL SOLUTION */
  russel(S, D);

  _EnterX = _EndX++;  /* AN EMPTY SLOT (ONLY _n1+_n2-1 BASIC VARIABLES) */

  return sSum > dSum ? dSum : sSum;
}


/**********************
    findBasicVariables
 **********************/
static void findBasicVariables(node1_t *U, node1_t *V)
{
  int i, j, found;
  int UfoundNum, VfoundNum;
  node1_t u0Head, u1Head, *CurU, *PrevU;
  node1_t v0Head, v1Head, *CurV, *PrevV;

  /* INITIALIZE THE ROWS LIST (U) AND THE COLUMNS LIST (V) */
  u0Head.Next = CurU = U;
  for (i=0; i < _n1; i++)
    {
      CurU->i = i;
      CurU->Next = CurU+1;
      CurU++;
    }
  (--CurU)->Next = NULL;
  u1Head.Next = NULL;

  CurV = V+1;
  v0Head.Next = _n2 > 1 ? V+1 : NULL;
  for (j=1; j < _n2; j++)
    {
      CurV->i = j;
      CurV->Next = CurV+1;
      CurV++;
    }
  (--CurV)->Next = NULL;
  v1Head.Next = NULL;

  /* THERE ARE _n1+_n2 VARIABLES BUT ONLY _n1+_n2-1 INDEPENDENT EQUATIONS,
     SO SET V[0]=0 */
  V[0].i = 0;
  V[0].val = 0;
  v1Head.Next = V;
  v1Head.Next->Next = NULL;

  /* LOOP UNTIL ALL VARIABLES ARE FOUND */
  UfoundNum=VfoundNum=0;
  while (UfoundNum < _n1 || VfoundNum < _n2)
    {

#if DEBUG_LEVEL > 3
      printf("UfoundNum=%d/%d,VfoundNum=%d/%d/n",UfoundNum,_n1,VfoundNum,_n2);
      printf("U0=");
      for(CurU = u0Head.Next; CurU != NULL; CurU = CurU->Next)
	printf("[%d]",CurU-U);
      printf("/n");
      printf("U1=");
      for(CurU = u1Head.Next; CurU != NULL; CurU = CurU->Next)
	printf("[%d]",CurU-U);
      printf("/n");
      printf("V0=");
      for(CurV = v0Head.Next; CurV != NULL; CurV = CurV->Next)
	printf("[%d]",CurV-V);
      printf("/n");
      printf("V1=");
      for(CurV = v1Head.Next; CurV != NULL; CurV = CurV->Next)
	printf("[%d]",CurV-V);
      printf("/n/n");
#endif
      
      found = 0;
      if (VfoundNum < _n2)
	{
	  /* LOOP OVER ALL MARKED COLUMNS */
	  PrevV = &v1Head;
	  for (CurV=v1Head.Next; CurV != NULL; CurV=CurV->Next)
	    {
	      j = CurV->i;
	      /* FIND THE VARIABLES IN COLUMN j */
	      PrevU = &u0Head;
	      for (CurU=u0Head.Next; CurU != NULL; CurU=CurU->Next)
		{
		  i = CurU->i;
		  if (_IsX[i][j])
		    {
		      /* COMPUTE U[i] */
		      CurU->val = _C[i][j] - CurV->val;
		      /* ...AND ADD IT TO THE MARKED LIST */
		      PrevU->Next = CurU->Next;
		      CurU->Next = u1Head.Next != NULL ? u1Head.Next : NULL;
		      u1Head.Next = CurU;
		      CurU = PrevU;
		    }
		  else
		    PrevU = CurU;
		}
	      PrevV->Next = CurV->Next;
	      VfoundNum++;
	      found = 1;
	    }
	}
     if (UfoundNum < _n1)
	{
	  /* LOOP OVER ALL MARKED ROWS */
	  PrevU = &u1Head;
	  for (CurU=u1Head.Next; CurU != NULL; CurU=CurU->Next)
	    {
	      i = CurU->i;
	      /* FIND THE VARIABLES IN ROWS i */
	      PrevV = &v0Head;
	      for (CurV=v0Head.Next; CurV != NULL; CurV=CurV->Next)
		{
		  j = CurV->i;
		  if (_IsX[i][j])
		    {
		      /* COMPUTE V[j] */
		      CurV->val = _C[i][j] - CurU->val;
		      /* ...AND ADD IT TO THE MARKED LIST */
		      PrevV->Next = CurV->Next;
		      CurV->Next = v1Head.Next != NULL ? v1Head.Next: NULL;
		      v1Head.Next = CurV;
		      CurV = PrevV;
		    }
		  else
		    PrevV = CurV;
		}
	      PrevU->Next = CurU->Next;
	      UfoundNum++;
	      found = 1;
	    }
	}
     if (! found)
       {
	 fprintf(stderr, "emd: Unexpected error in findBasicVariables!/n");
	 fprintf(stderr, "This typically happens when the EPSILON defined in/n");
	 fprintf(stderr, "emd.h is not right for the scale of the problem./n");
	 exit(1);
       }
    }
}




/**********************
    isOptimal
 **********************/
static int isOptimal(node1_t *U, node1_t *V)
{    
  double delta, deltaMin;
  int i, j, minI, minJ;

  /* FIND THE MINIMAL Cij-Ui-Vj OVER ALL i,j */
  deltaMin = INFINITY;
  for(i=0; i < _n1; i++)
    for(j=0; j < _n2; j++)
      if (! _IsX[i][j])
	{
	  delta = _C[i][j] - U[i].val - V[j].val;
	  if (deltaMin > delta)
	    {
              deltaMin = delta;
	      minI = i;
	      minJ = j;
	    }
	}

#if DEBUG_LEVEL > 3
  printf("deltaMin=%f/n", deltaMin);
#endif

   if (deltaMin == INFINITY)
     {
       fprintf(stderr, "emd: Unexpected error in isOptimal./n");
       exit(0);
     }
   
   _EnterX->i = minI;
   _EnterX->j = minJ;
   
   /* IF NO NEGATIVE deltaMin, WE FOUND THE OPTIMAL SOLUTION */
   return deltaMin >= -EPSILON * _maxC;

/*
   return deltaMin >= -EPSILON;
 */
}


/**********************
    newSol
**********************/
static void newSol()
{
    int i, j, k;
    double xMin;
    int steps;
    node2_t *Loop[2*MAX_SIG_SIZE1], *CurX, *LeaveX;
 
#if DEBUG_LEVEL > 3
    printf("EnterX = (%d,%d)/n", _EnterX->i, _EnterX->j);
#endif

    /* ENTER THE NEW BASIC VARIABLE */
    i = _EnterX->i;
    j = _EnterX->j;
    _IsX[i][j] = 1;
    _EnterX->NextC = _RowsX[i];
    _EnterX->NextR = _ColsX[j];
    _EnterX->val = 0;
    _RowsX[i] = _EnterX;
    _ColsX[j] = _EnterX;

    /* FIND A CHAIN REACTION */
    steps = findLoop(Loop);

    /* FIND THE LARGEST VALUE IN THE LOOP */
    xMin = INFINITY;
    for (k=1; k < steps; k+=2)
      {
	if (Loop[k]->val < xMin)
	  {
	    LeaveX = Loop[k];
	    xMin = Loop[k]->val;
	  }
      }

    /* UPDATE THE LOOP */
    for (k=0; k < steps; k+=2)
      {
	Loop[k]->val += xMin;
	Loop[k+1]->val -= xMin;
      }

#if DEBUG_LEVEL > 3
    printf("LeaveX = (%d,%d)/n", LeaveX->i, LeaveX->j);
#endif

    /* REMOVE THE LEAVING BASIC VARIABLE */
    i = LeaveX->i;
    j = LeaveX->j;
    _IsX[i][j] = 0;
    if (_RowsX[i] == LeaveX)
      _RowsX[i] = LeaveX->NextC;
    else
      for (CurX=_RowsX[i]; CurX != NULL; CurX = CurX->NextC)
	if (CurX->NextC == LeaveX)
	  {
	    CurX->NextC = CurX->NextC->NextC;
	    break;
	  }
    if (_ColsX[j] == LeaveX)
      _ColsX[j] = LeaveX->NextR;
    else
      for (CurX=_ColsX[j]; CurX != NULL; CurX = CurX->NextR)
	if (CurX->NextR == LeaveX)
	  {
	    CurX->NextR = CurX->NextR->NextR;
	    break;
	  }

    /* SET _EnterX TO BE THE NEW EMPTY SLOT */
    _EnterX = LeaveX;
}



/**********************
    findLoop
**********************/
static int findLoop(node2_t **Loop)
{
  int i, steps;
  node2_t **CurX, *NewX;
  char IsUsed[2*MAX_SIG_SIZE1]; 
 
  for (i=0; i < _n1+_n2; i++)
    IsUsed[i] = 0;

  CurX = Loop;
  NewX = *CurX = _EnterX;
  IsUsed[_EnterX-_X] = 1;
  steps = 1;

  do
    {
      if (steps%2 == 1)
	{
	  /* FIND AN UNUSED X IN THE ROW */
	  NewX = _RowsX[NewX->i];
	  while (NewX != NULL && IsUsed[NewX-_X])
	    NewX = NewX->NextC;
	}
      else
	{
	  /* FIND AN UNUSED X IN THE COLUMN, OR THE ENTERING X */
	  NewX = _ColsX[NewX->j];
	  while (NewX != NULL && IsUsed[NewX-_X] && NewX != _EnterX)
	    NewX = NewX->NextR;
	  if (NewX == _EnterX)
	    break;
 	}

     if (NewX != NULL)  /* FOUND THE NEXT X */
       {
	 /* ADD X TO THE LOOP */
	 *++CurX = NewX;
	 IsUsed[NewX-_X] = 1;
	 steps++;
#if DEBUG_LEVEL > 3
	 printf("steps=%d, NewX=(%d,%d)/n", steps, NewX->i, NewX->j);    
#endif
       }
     else  /* DIDN'T FIND THE NEXT X */
       {
	 /* BACKTRACK */
	 do
	   {
	     NewX = *CurX;
	     do 
	       {
		 if (steps%2 == 1)
		   NewX = NewX->NextR;
		 else
		   NewX = NewX->NextC;
	       } while (NewX != NULL && IsUsed[NewX-_X]);
	     
	     if (NewX == NULL)
	       {
		 IsUsed[*CurX-_X] = 0;
		 CurX--;
		 steps--;
	       }
	 } while (NewX == NULL && CurX >= Loop);
	 
#if DEBUG_LEVEL > 3
	 printf("BACKTRACKING TO: steps=%d, NewX=(%d,%d)/n",
		steps, NewX->i, NewX->j);    
#endif
           IsUsed[*CurX-_X] = 0;
	   *CurX = NewX;
	   IsUsed[NewX-_X] = 1;
       }     
    } while(CurX >= Loop);
  
  if (CurX == Loop)
    {
      fprintf(stderr, "emd: Unexpected error in findLoop!/n");
      exit(1);
    }
#if DEBUG_LEVEL > 3
  printf("FOUND LOOP:/n");
  for (i=0; i < steps; i++)
    printf("%d: (%d,%d)/n", i, Loop[i]->i, Loop[i]->j);
#endif

  return steps;
}



/**********************
    russel
**********************/
static void russel(double *S, double *D)
{
  int i, j, found, minI, minJ;
  double deltaMin, oldVal, diff;
  double Delta[MAX_SIG_SIZE1][MAX_SIG_SIZE1];
  node1_t Ur[MAX_SIG_SIZE1], Vr[MAX_SIG_SIZE1];
  node1_t uHead, *CurU, *PrevU;
  node1_t vHead, *CurV, *PrevV;
  node1_t *PrevUMinI, *PrevVMinJ, *Remember;

  /* INITIALIZE THE ROWS LIST (Ur), AND THE COLUMNS LIST (Vr) */
  uHead.Next = CurU = Ur;
  for (i=0; i < _n1; i++)
    {
      CurU->i = i;
      CurU->val = -INFINITY;
      CurU->Next = CurU+1;
      CurU++;
    }
  (--CurU)->Next = NULL;
  
  vHead.Next = CurV = Vr;
  for (j=0; j < _n2; j++)
    {
      CurV->i = j;
      CurV->val = -INFINITY;
      CurV->Next = CurV+1;
      CurV++;
    }
  (--CurV)->Next = NULL;
  
  /* FIND THE MAXIMUM ROW AND COLUMN VALUES (Ur[i] AND Vr[j]) */
  for(i=0; i < _n1 ; i++)
    for(j=0; j < _n2 ; j++)
      {
	float v;
	v = _C[i][j];
	if (Ur[i].val <= v)
	  Ur[i].val = v;
	if (Vr[j].val <= v)
	  Vr[j].val = v;
      }
  
  /* COMPUTE THE Delta MATRIX */
  for(i=0; i < _n1 ; i++)
    for(j=0; j < _n2 ; j++)
      Delta[i][j] = _C[i][j] - Ur[i].val - Vr[j].val;

  /* FIND THE BASIC VARIABLES */
  do
    {
#if DEBUG_LEVEL > 3
      printf("Ur=");
      for(CurU = uHead.Next; CurU != NULL; CurU = CurU->Next)
	printf("[%d]",CurU-Ur);
      printf("/n");
      printf("Vr=");
      for(CurV = vHead.Next; CurV != NULL; CurV = CurV->Next)
	printf("[%d]",CurV-Vr);
      printf("/n");
      printf("/n/n");
#endif
 
      /* FIND THE SMALLEST Delta[i][j] */
      found = 0; 
      deltaMin = INFINITY;      
      PrevU = &uHead;
      for (CurU=uHead.Next; CurU != NULL; CurU=CurU->Next)
	{
	  int i;
	  i = CurU->i;
	  PrevV = &vHead;
	  for (CurV=vHead.Next; CurV != NULL; CurV=CurV->Next)
	    {
	      int j;
	      j = CurV->i;
	      if (deltaMin > Delta[i][j])
		{
		  deltaMin = Delta[i][j];
		  minI = i;
		  minJ = j;
		  PrevUMinI = PrevU;
		  PrevVMinJ = PrevV;
		  found = 1;
		}
	      PrevV = CurV;
	    }
	  PrevU = CurU;
	}
      
      if (! found)
	break;

      /* ADD X[minI][minJ] TO THE BASIS, AND ADJUST SUPPLIES AND COST */
      Remember = PrevUMinI->Next;
      addBasicVariable(minI, minJ, S, D, PrevUMinI, PrevVMinJ, &uHead);

      /* UPDATE THE NECESSARY Delta[][] */
      if (Remember == PrevUMinI->Next)  /* LINE minI WAS DELETED */
	{
	  for (CurV=vHead.Next; CurV != NULL; CurV=CurV->Next)
	    {
	      int j;
	      j = CurV->i;
	      if (CurV->val == _C[minI][j])  /* COLUMN j NEEDS UPDATING */
		{
		  /* FIND THE NEW MAXIMUM VALUE IN THE COLUMN */
		  oldVal = CurV->val;
		  CurV->val = -INFINITY;
		  for (CurU=uHead.Next; CurU != NULL; CurU=CurU->Next)
		    {
		      int i;
		      i = CurU->i;
		      if (CurV->val <= _C[i][j])
			CurV->val = _C[i][j];
		    }
		  
		  /* IF NEEDED, ADJUST THE RELEVANT Delta[*][j] */
		  diff = oldVal - CurV->val;
		  if (fabs(diff) < EPSILON * _maxC)
		    for (CurU=uHead.Next; CurU != NULL; CurU=CurU->Next)
		      Delta[CurU->i][j] += diff;
		}
	    }
	}
      else  /* COLUMN minJ WAS DELETED */
	{
	  for (CurU=uHead.Next; CurU != NULL; CurU=CurU->Next)
	    {
	      int i;
	      i = CurU->i;
	      if (CurU->val == _C[i][minJ])  /* ROW i NEEDS UPDATING */
		{
		  /* FIND THE NEW MAXIMUM VALUE IN THE ROW */
		  oldVal = CurU->val;
		  CurU->val = -INFINITY;
		  for (CurV=vHead.Next; CurV != NULL; CurV=CurV->Next)
		    {
		      int j;
		      j = CurV->i;
		      if(CurU->val <= _C[i][j])
			CurU->val = _C[i][j];
		    }
		  
		  /* If NEEDED, ADJUST THE RELEVANT Delta[i][*] */
		  diff = oldVal - CurU->val;
		  if (fabs(diff) < EPSILON * _maxC)
		    for (CurV=vHead.Next; CurV != NULL; CurV=CurV->Next)
		      Delta[i][CurV->i] += diff;
		}
	    }
	}
    } while (uHead.Next != NULL || vHead.Next != NULL);
}




/**********************
    addBasicVariable
**********************/
static void addBasicVariable(int minI, int minJ, double *S, double *D, 
			     node1_t *PrevUMinI, node1_t *PrevVMinJ,
			     node1_t *UHead)
{
  double T;
  
  if (fabs(S[minI]-D[minJ]) <= EPSILON * _maxW)  /* DEGENERATE CASE */
    {
      T = S[minI];
      S[minI] = 0;
      D[minJ] -= T; 
    }
  else if (S[minI] < D[minJ])  /* SUPPLY EXHAUSTED */
    {
      T = S[minI];
      S[minI] = 0;
      D[minJ] -= T; 
    }
  else  /* DEMAND EXHAUSTED */
    {
      T = D[minJ];
      D[minJ] = 0; 
      S[minI] -= T; 
    }

  /* X(minI,minJ) IS A BASIC VARIABLE */
  _IsX[minI][minJ] = 1; 

  _EndX->val = T;
  _EndX->i = minI;
  _EndX->j = minJ;
  _EndX->NextC = _RowsX[minI];
  _EndX->NextR = _ColsX[minJ];
  _RowsX[minI] = _EndX;
  _ColsX[minJ] = _EndX;
  _EndX++;

  /* DELETE SUPPLY ROW ONLY IF THE EMPTY, AND IF NOT LAST ROW */
  if (S[minI] == 0 && UHead->Next->Next != NULL)
    PrevUMinI->Next = PrevUMinI->Next->Next;  /* REMOVE ROW FROM LIST */
  else
    PrevVMinJ->Next = PrevVMinJ->Next->Next;  /* REMOVE COLUMN FROM LIST */
}





/**********************
    printSolution
**********************/
static void printSolution()
{
  node2_t *P;
  double totalCost;

  totalCost = 0;

#if DEBUG_LEVEL > 2
  printf("SIG1/tSIG2/tFLOW/tCOST/n");
#endif
  for(P=_X; P < _EndX; P++)
    if (P != _EnterX && _IsX[P->i][P->j])
      {
#if DEBUG_LEVEL > 2
	printf("%d/t%d/t%f/t%f/n", P->i, P->j, P->val, _C[P->i][P->j]);
#endif
	totalCost += (double)P->val * _C[P->i][P->j];
      }

  printf("COST = %f/n", totalCost);
}


#ifndef _EMD_H
#define _EMD_H
/*
    emd.h

    Last update: 3/24/98

    An implementation of the Earth Movers Distance.
    Based of the solution for the Transportation problem as described in
    "Introduction to Mathematical Programming" by F. S. Hillier and 
    G. J. Lieberman, McGraw-Hill, 1990.

    Copyright (C) 1998 Yossi Rubner
    Computer Science Department, Stanford University
    E-Mail: rubner@cs.stanford.edu   URL: http://vision.stanford.edu/~rubner
*/


/* DEFINITIONS */
#define MAX_SIG_SIZE   100
#define MAX_ITERATIONS 500
#define INFINITY       1e20
#define EPSILON        1e-6

/*****************************************************************************/
/* feature_t SHOULD BE MODIFIED BY THE USER TO REFLECT THE FEATURE TYPE      */
typedef int feature_t;
/*****************************************************************************/


typedef struct
{
  int n;                /* Number of features in the signature */
  feature_t *Features;  /* Pointer to the features vector */
  float *Weights;       /* Pointer to the weights of the features */
} signature_t;


typedef struct
{
  int from;             /* Feature number in signature 1 */
  int to;               /* Feature number in signature 2 */
  float amount;         /* Amount of flow from "from" to "to" */
} flow_t;



float emd(signature_t *Signature1, signature_t *Signature2,
	  float (*func)(feature_t *, feature_t *),
	  flow_t *Flow, int *FlowSize);

#endif

  
Code for the Earth Movers Distance (EMD) 
  
 
Introduction
This is an implementation of the Earth Movers Distance, as described in [1]. The EMD computes the distance between two distributions, which are represented by signatures. The signatures are sets of weighted features that capture the distributions. The features can be of any type and in any number of dimensions, and are defined by the user. 
The EMD is defined as the minimum amount of work needed to change one signature into the other. The notion of "work" is based on the user-defined ground distance which is the distance between two features. The size of the two signatures can be different. Also, the sum of weights of one signature can be different than the sum of weights of the other (partial match). Because of this, the EMD is normalized by the smaller sum. 
The code is implemented in C, and is based on the solution for the Transportation problem as described in [2] 
Please let me know of any bugs you find, or any questions, comments, suggestions, and criticisms you have. If you find this code useful for your work, I would like very much to hear from you. Once you do, I'll inform you of any improvements, etc. Also, an acknowledgment in any publication describing work that uses this code would be greatly appreciated. 
 
Usage
To use the code, perform the following steps: 
      
download the files emd.h and emd.c (check the log of changes for latest changes). 
In emd.h, modify the line 
typedef int feature_t;
to reflect your feature data type. Structures may be used too, for example 
typedef struct {
   int X,Y,Z;
} feature_t;
To compute an EMD, call: 
float emd(signature_t *Signature1, signature_t *Signature2,
	  float (*func)(feature_t *, feature_t *),
	  flow_t *Flow, int *FlowSize);
where 
        

         
          Signature1, Signature2: 
         
         

          Pointers to the two signatures that their distance we want to compute. 
         
         
          Dist: 
         
         

          Pointer to the ground distance function. i.e. the function that computes the distance between two features. 
         
         
          Flow: 
         
         

          (Optional) Pointer to a vector of flow_t (defined in emd.h) where the resulting flow will be stored. Flow must have n1+n2-1 elements, where n1 and n2 are the sizes of the two signatures respectively. If 
          NULL, the flow is not returned. 
         
         
          FlowSize: 
         
         

          (Optional) In case 
          Flow is not 
          NULL, 
          FlowSize should point to a integer where the number of flow elements (always less or equal to n1+n2-1) will be written. 
         
        
Compile emd.c and link it to your code. 
The signature data type signature_t is defined in emd.h as: 
typedef struct
{
  int n;                /* Number of features in the distribution */
  feature_t *Features;  /* Pointer to the features vector */
  float *Weights;       /* Pointer to the weights of the features */
} signature_t;
The feature data type feature_t is defined in emd.h and should be modified by the user to reflect his feature type. 
In special cases, the user might want to change some of the values in the definitions in emd.h: 
      

       
        #define MAX_SIG_SIZE 100 
       
       

        The maximum allowed number of features in a signature 
       
       
        #define MAX_ITERATIONS 100 
       
       

        Maximum number of iterations. For ordinary problems, 100 should be more than enough. 
       
       
        #define INFINITY 1e20 
       
       

        INFINITY should be much larger than any other value in the problem 
       
       
        #define EPSILON 1e-6 
       
       

        EPSILON determines the accuracy of the solution. 
       
       
          
       
      
Examples

      
example1.c
In this example the features are in three-dimensional space, and the ground distance is the Euclidean distance.
To try this example, you need to modify feature_t in emd.h to be 

typedef struct { int X,Y,Z; } feature_t;
example2.c
Here, instead of providing a function to compute the ground distances, they are given in a predefined matrix. This is done by defining feature_t as int (the default), and setting the features in each signature to have consecutive numbers. The ground distance function uses these numbers as indexes to the predefined cost matrix.
The resulting flow is returned from the emd function by passing as the last parameters a vector of flow_t, and a pointer to int where the actual number of flow elements will be written.
Also, in this example the total weights of the two signatures are different. The first signature has a total weight of 1 while the second signature has a total weight of 0.9. 
 
Log of changes

      
Date
Decription
05/10/98
Changed from absolute error check to relative error check + Fixed a bug that caused the "Unexpected error in findBasicVariables" error message (thanks Mark Ruzon) 
03/04/98
Fixed a bug that sometime caused a crash when the second signature had only one feature (thanks Mark Ruzon) 
03/31/98
Fixed a bug in the case where *both* signatures have only one feature (thanks Rajesh Batra) 

References
[1] Y. Rubner, C. Tomasi, and L. J. Guibas. A Metric for Distributions with Applications to Image Databases. Proceedings of the 1998 IEEE International Conference on Computer Vision, Bombay, India, January 1998, pp. 59-66.

[2] F. S. Hillier and G. J. Lieberman. Introduction to Mathematical Programming McGraw-Hill, 1990.