A patient presented with local gigantism of the leg and increased pulsations of the lower limb veins. Most probable diagnosis is :
The patient is having congenital AV fistula which has caused overgrowth of the leg.
Arteriovenous fistula (AVF) 

AVF may be congenital or acquired (by the trauma of a penetrating wound, or surgically created for renal dialysis)
Structural effect on veins → Veins are arterialised*, they become dilated tortuous and thick walled.
Physiological effects of fistula -
Increased pulse pressure (Increased systolic & decreased diastolic)
Enhanced venous return result in Increased HR* and Increased CO*.
Left ventricular enlargement and later cardiac failure may occur
A congenital fistula in the young may cause overgrowth of limb*.
In the leg, indolent ulcers may result* from relative ischemia below the short circuit.

Clinical signs

A pulsatile swelling*
A thrill on palpation*
Continuous bruit on auscultation*
Nicoladonis sign' or Branham sign* → Pressure on the artery proximal to fistula causes the swelling to diminish in size, a thrill or bruit to cease, the pulse rate to fall and the pulse pressure return to normal.
Tumor
AV fistula
Varicose veins
Incompetence of the saphenofemoral junction